'::ji^- 


nm 


Jh'y. 


..i?^. 


AN 


INTRODUCTION 


TO 


iiIL(Bl^IBIE^ 


9 


fiEIl^TG   THE 


FIRST     PART 


OE  A 


COURSE  OF  MATHEMATICS, 

ADAPTED  TO 

THE  METHOD  OF  INSTRUCTION  IN  THE   HIGHER  SCHOOLS  AND 
ACADEMIES  IN  THE  UNITED  STATES. 


By  JEREMIAH  DAY,  ll.d. 

President  of  Yale  College. 


PUBLISHED  BY  HOWE  &  SPALDINO. 


PLAGG  &  GRAY,  PRINTERS. 
1819. 


2.i-A*4^3 


OAJS: 
^37 


THE  following  elementary  compendium  is  the  same  as 
the  Introduction  to  Algebra  intended  for  the  use  of  Col- 
leges, with  the  omission  of  a  few  of  the  sections  at  the 
latter  end  of  the  book.  These  it  was  supposed  would 
not  generally  be  needed,  in  private  schools  and  academies. 
The  parts  retained  contain  the  elements  of  the  science. 
The  first  two  hundred  pages  are  precisely  the  same  as  in 
the  second  edition  of  the  larger  work.  The  subjects  in 
the  first  edition,  which  are  omitted  in  this  abridgment,  are 
Variation,  Progression,  Infinity,  the  Binomial  Theorem, 
Resolution  of  the  higher  Equations,  AppUcation  of  Al- 
gebra to  Geometry,  and  Equations  of  Curves. 


OF  THE 


;uinvEES 


TY 


CO^'TEJfTS. 


Paor 
IntroductoFy  Observations,  on  the  Mathematics  in 

genera),  1 

ALGEBRA. 

Section  I.  Notation,  Positive  and  Negative  quan- 

ties,  Axioms,  &c.  8 

II.  Addition,  21 

III.  Subtraction,  27 
rV.  Multiphcation,  31 
V.  Division,  41 
Vr.  Algebraic  Fractions,  49 

VII.  Reduction  of  Equations,  by  Transposi- 

tion, Multiplication,  and   Division. 
Solution  of  Problems,  65 

VIII.  Involution.     Notation,  addition,  sub- 

traction, multiphcation,  and  division 

of  Powers,  83 

IX.  Evolution.     Notation,  reduction,  addi- 

tion, subtraction,  multiphcation,  di-    . 
vision,  and   involution,   of    Radical 
Quantities,  99 

X.  Reduction  of  Equations  by  Involution 

and  Evolution.     Affected  Quadratic 
Equations,  128 

XI.  Solution  of  Problems   which   contain 

two  or  more  Unknown  Quantities,  153 

XII.  Ratio  and  Proportion,  173 

XIII.  Division  by  Compound  Divisors,  205 

XIV.  Evolution  of  Compound  Quantities,  210 

XV.  Infinite  Series,  214 


INTRODUCTORY  OBSERVATIONS 


OW   THE 


MkatTaeiaciatics  in  Gfeneral, 


Art.  1.  Mathematics  is  the  science  of  quantity. 

Any  thing  which  can  be  multiplied,  divided,  or  measured,  is 
called  quantity.  Thus,  a  line  is  a  quantity,  because  it  can  be 
doubled,  trebled,  or  halved  ;  and  can  be  measured,  by  apply- 
ing to  it  another  line,  as  a  foot,  a  yard,  or  an  ell.  Weight  is 
a  quantity,  which  can  be  measured,  in  pounds,  ounces,  and 
grains.  i\me  is  a  species  of  quantity,  whose  measure  can 
be  expressed,  in  hours,  minutes,  and  seconds.  But  colour  is 
not  a  quantity.  It  cannot  be  said,  with  propriety,  that  one 
colour  is  twice  as  great,  or  half  as  great  as  another.  The 
operations  of  the  mind,  such  as  thought,  choice,  desire,  ha- 
ired, Szc.  are  not  quantities.  They  are  incapable  of  mensu- 
ration.* 

2.  Those  parts  of  the  Mathematics,  on  which  all  the  othere 
are  founded,  are  Arithmetic,  Algebra,  and  Geometry, 

3.  Arithmetic  is  the  science  of  numbers.  Its  aid  is  re- 
quired, to  complete  and  apply  the  calculations,  in  almost 
every  other  department  of  the  mathematics. 

4.  Algebra  is  a  method  af  computing  by  letters  and  other 
symbols.  Fluxions,  or  the  Differential  and  Integral  Calcu- 
lus, may  be  considered  as  belonging  to  the  higher  branches  of 
algebra,  t 

5.  Geometry  is  that  part  of  the  mathematics,  which  treats 
of  magnitude.     By  magnitude,  in  the  appropriate  sense  of 

*  See  note  A.  t  See  note  B, 

o 


:j  mathematics. 

tlie  term,  is  meant  that  species  of  quantity,  which  is  extend- 
(d  ;  that  is,  which  has  one  or  more  of  the  three  dimensions, 
length,  breadth,  and  thickness.  Thus  a  line  is  a  magnitude, 
because  it  is  extended,  in  length.  A  surface  is  a  magnitude, 
liaving  length  and  breadth.  A  solid  is  a  magnitude,  having 
lengtli,  breadth,  and  thickness.  But  motion,  though  a  quan- 
tity, is  not,  strictly  speaking,  a  magnitude.  It  has:  neither 
length,  breadth,  nor  thickness.*  ' 

6.  Trigonometry  and  Conic  Sections  are  branches  of 
the  mathematics,  in  which,  the  principles  of  geometry  are 
applied  to  triangles,  and  the  sections  of  a  cone, 

7.  Mathematics  are  either  pure,  or  mixed,  \npure  math- 
ematics, quantities-  are  considered,  independently  of  any  sub- 
stances actually  existing.  But,  in  mixed  mathematics,  the 
relations  of  quantities  are  investigated,  in  connection  with 
some  of  the  properties  of  matter,  or  with  reference  to  the 
common  transactions  of  business.  Thus,  in  Surveying, 
mathematical  principles  are  applied  to  the  m.ea£uring  of 
land  ;  in  Optics,  to  the  properties  of  light ;  and  in  Astronomy, 
to  the  motions  of  the  heavenly  bodies. 

8.  The  science  of  the  pure  mathematics  has  long  been 
distinguished,  for  the  clearness  and  distinctness  of  its  princi- 
ples ;  and  the  irresistible  conviction,  which  they  carry  to  the 
mind  of  every  one  who  is  once  made  acquainted  with  them. 
This  is  to  be  ascribed,  partly  to  the  nature  of  the  subjects, 
and  partly  to  the  exactness  of  the  definitions,  the  axioms,  and 
the  demonstrations. 

9.  The  foundation  of  all  mathematical  knowledge  must  be 
laid  in  detinitions.  A  definition  is  an  explanation  of  what  is 
meant,  by  any  word  or  phrase.  Thus,  an  equilateral  triangle 
is  defined,  by  saying,  that  it  is  a  figure  bounded  by  three 
equal  sides. 

It  is  essential  to  a  complete  definition,  that  it  perfectly  dis- 
tinguish the  thing  defined,  from  every  thing  else.  On  many 
subjects,  it  is  difllicult  to  give  such  precision  to  language,  that 
it  shall  convey,  to  every  hearer  or  reader,  exactly  the  same 
ideas.  But,  in  the  mathematics,  the  principal  terms  may  be 
so  defined,  as  not  to  leave  room  for  the  least  difference  of 
apprehension,  respecting  their  meaning.  All  must  be  agreed^ 
as  to  the  nature  of  a  circle,  a  square,  and  a  triangle,  when 
they  have  once  learned  the  definitions  of  these  figures. 

■''•'  Some  writers,  however,  use  magnitude,  as  synonymous  with  quantity. 


MATHEMATICS.  3 

Under  the  head  of  definitions,  may  be  inchided  explana- 
tions of  the  characters  which  are  used  to  denote  the  relations 
ei  quantities.  Thus,  the  character  -/  is  explained  or  defined, 
by  saying  that  it  signifies  the  same  as  the  words  square  root. 

10.  The  next  step,  after  becoming  acquainted  with  the 
meaning  of  mathematical  terms,  is  to  bring  them  together,  in 
the  form  of  propositions.  Some  of  the  relations  of  quanti- 
ties require  no  process  of  reasoning,  to  render  them  evident. 
To  be  understood,  they  need  only  to  be  proposed.  That  a 
square  is  a  different  figure  from  a  circle  ;  that  the  whole  of  a 
thing  is  greater,  than  one  of  its  parts  ;  and,  that  two  straight 
hues  cannot  inclose  a  space,  are  propositions  so  manifestly 
true,  that  no  reasoning  upon  them  could  make  them  more 
certain.  They  are,  therefore,  called  self-evident  truths,  or 
axioms. 

1 1 .  There  are,  however,  comparatively  few  mathematical 
truths  which  are  self-evident.  Most  require  to  be  proved,  by 
a  chain  of  reasoning.  Propositions  of  this  nature  are  de- 
nominated theorems  ^  and  the  process,  by  which,  they  arv^ 
shown  to  be  true,  is  called  demonstration.  This  is  a  mode  of 
arguing,  in  which,  every  inference  is  immediately  derived, 
either  from  definitions,  or  from  principles  which  have  been 
previously  demonstrated.  In  this  way,  complete  certainty  is 
made  to  accompany  every  step,  in  a  long  course  of  reasoning. 

12.  Demonstration  is  either  direct,  or  indirect.  The  for- 
mer is  the  common,  obvious  mode  of  conducting  a  demon- 
strative argument.  But,  in  some  instances,  it  is  necessary  to 
resort  to  indirect  demonstration  ;  which  is  a  method  of  estab- 
lishing a  proposition,  hy  proving  that  to  suppose  it  not  true, 
w^ould  lead  to  an  absurdity.  This  is  frequently  called  reduc- 
tio  ad  ahsurdum.  Thus,  in  certain  cases  in  geometry,  two 
lines  may  be  proved  to  be  equal,  by  shewing  that  to  suppose 
them  unequal,  would  involve  an  absurdity. 

1 3.  Besides  the  principal  theorems  in  the  mathematics, 
there  are  also  Lemmas,  and  Corollaries,  A  Lemma  is  a  pro- 
position which  is  demonstrated,  for  the  purpose  of  using  it,  in 
the  demonstration  of  «ome  other  proposition.  This  prepa- 
ratory step  is  taken  to  prevent  the  proof  of  the  principal 
theorem  from  becoming  comphcated  and  tedious. 

14.  A  Corollary  is  an  inference  from  a  preceding  proposi- 
tion. A  Scholium  is  a  remark  of  any  kind,  suggested  by 
something  which  has  gone  before,  though  not,  like  a  corollary, 
immediately  depending  on  it. 


4  MATHEMATICS. 

15.  The  immediate  object  of  inquiry,  in  the  mathematics, 
is,  frequently,  not  the  demonstration  of  a  general  truth,  but  a 
method  of  performing  some  operation,  such  as  reducing  a 
vulgar  fraction  to  a  decimal,  extracting  the  cube  root,  or  in- 
scribing a  circle  in  a  square.  This  is  called  solving  a  pro- 
blem. A  theorem  is  something  to  be  proved.  A  problem  is 
something  to  be  done, 

16.  When  that  which  is  required  to  be  done,  is  so  easy,  as 
to  be  obvious  to  every  one,  without  an  explanation,  it  is  call- 
ed a  postulate.  Of  this  nature,  is  the  drawing  of  a  straiglit 
line,  from  one  point  to  another. 

1 7.  A  quantity  is  said  to  be  given,  when  it  is  either  sup- 
posed to  be  already  known,  or  is  made  a  condition,  in  the 
statement  of  any  theorem  or  problem.  In  the  rule  of  pro- 
portion in  arithmetic,  for  instance,  three  terms  must  be  giv- 
en, to  enable  us  to  find  a  fourth.  These  three  terms  are  the 
data,  upon  which  the  calculation  is  founded.  If  we  are  re- 
quired to  find  the  number  of  acres,  in  a  circular  island  ten 
miles  in  circumference,  the  circular  figure,  and  the  length  of 
the  circumference,  are  the  data.  They  are  said  to  be  given 
hy  supposition,  that  is,  by  the  conditions  of  the  problem.  A 
quantity  is  also  said  to  be  given,  when  it  may  be  directly  and 
easily  inferred,  from  something  else  which  is  given.  Thus, 
if  two  numbers  are  given,  their  sum  is  given  ;  because  it  is 
obtained,  by  merely  adding  the  numbers  together. 

In  Geometry,  a  quantity  may  be  given,  either  in  position^ 
or  magnitude,  or  both.  A  line  is  given  in  position,  when  its 
situation  and  direction  are  known.  It  is  given  in  magnitude, 
when  its  length  is  known.  A  circle  is  given  in  position,  wh^n 
the  place  of  its  centre  is  known.  It  is  given  in  magnitude^ 
when  the  length  of  its  diameter  is  known. 

18.  One  proposition  is  contra?!/,  or  contradictory  to  anotli- 
er,  when,  what  is  affirmed,  in  the  one,  is  denied,  in  the  other. 
A  proposition  and  its  contrary,  can  never  both  be  true.  It 
cannot  be  true,  that  two  given  lines  are  equal,  and  that  they 
are  not  equal,  at  the  same  time. 

1 9.  One  proposition  is  the  converse  of  another,  when  the 
order  is  inverted  ;  so  that,  what  is  given  or  supposed,  in  the 
first,  becomes  the  conclusion,  in  the  last ;  and  what  is  given 
in  the  last,  is  the  conclusion,  in  the  first.  Thus,  it  can  be 
proved,  first,  that  if  the  sides  of  a  triangle  are  equal,  the  an- 
gles are  equal  5  and  secondly,  that  if  the  angles  are  equal, 
the  sides  are  equal.    Here,  in  the  first  proposition,  the  equal- 


MATHEMATICS.  5 

lly  of  the  sides  is  given  :  and  tlie  equality  of  the  angles  in- 
ferred :  in  the  second,  the  equaUty  of  the  angles  is  given, 
.and  the  equahty  of  the  sides  inferred.  In  many  instances,  a 
proposition  and  its  converse  are  both  true  ;  as  in  the  prece- 
ding example.  But  this  is  not  always  the  case.  A  circle  is 
a  figure  bounded  by  a  curve  ;  but  a  figure  bounded  by  a  curve 
is  not  of  course  a  eircle. 

20.  The  practical  applications  of  the  mathematics,  in  the 
common  concerns  of  business,  in  the  useful  arts,  and  in  the 
various  branches  of  physical  science,  are  almost  innumerable. 
Mathematical  principles  are  necessary,  in  mercantile  transac- 
tions^ for  keeping,  arranging,  and  setthng  accounts,  adjusting 
the  prices  of  commodities,  and  calculating  the  profits  of  trade  ; 
in  Navigation^  for  directing  the  course  of  a  ship  on  the  ocean, 
adapting  the  position  of  her  sails  to  the  direction  of  the  wind, 
finding  her  latitude  and  longitude,  and  determining  the  bear- 
ings and  distances  of  objects  on  shore  :  in  Surveyings  for 
measuring,  dividing,  and  laying  out  grounds,  taking  the  eleva- 
tion of  hills,  and  fixing  the  boundaries  of  fields,  estates,  and 
public  territories  :  in  Mechanics^  for  understanding  the  laws 
of  motion,  the  composition  of  forces,  the  equilibrium  of  the 
mechanical  powers,  and  the  structure  of  machines  :  in  Archi- 
tecture, for  calculating  the  comparative  strength  of  timbers, 
the  pressure  which  each  will  be  required  to  sustain,  the  forms 
of  arches,  the  proportions  of  columns,  &;c. j  in  Fortification^ 
for  adjusting  the  position,  lines,  and  angles,  of  the  several 
parts  of  the  works  :  in  Gunnery^  for  regulating  the  elevation 
of  the  cannon,  the  force  of  the  powder,  and  the  velocity  and 
range  of  the  shot :  in  Optics,  for  tracing  the  direction  of  the 
rays  of  light,  understanding  the  formation  of  images,  the  laws 
of  vision,  the  separation  of  colours,  the  nature  of  the  rain- 
bow, and  the  construction  of  microscopes  and  telescopes  ;  in 
Astronomy,  for  computing  the  distances,  magnitudes,  and  re- 
volutions of  the  heavenly  bodies  ;  and  the  influence  of  the 
law  of  gravitation,  in  raising  the  tides,  disturbing  the  motions 
of  the  moon,  causing  the  return  of  the  comets,  and  retaining 
the  planets  in  their  orbits  :  in  Geography,  for  determining  the 
figure  and  dimensions  of  the  earth,  the  extent  of  oceans, 
islands,  continents,  and  countries ;  the  latitude  and  longitude 
of  places,  the  courses  of  rivers,  the  height  of  mountains,  and 
the  boundaries  of  kingdoms  :  in  History,  for  fixing  the  chro- 
nology of  remarkable  events,  and  estimating  the  strength  of 
armies,  the  wealth  of  nations,  the  value  of  their  revenues, 
and  the  amount  of  their  population  :  and^  in  the  concerns  of 


6  MATHEMATICS, 

Government^  for  apportioning  taxes,  arranging  schemes  of 
tinance,  and  regulating  national  expenses.  The  mathematics 
have  also  important  applications  to  Chemistry,  Mineralogy, 
Music,  Painting,  Sculpture,  and  indeed  to  a  great  proportion 
of  the  whole  circle  of  arts  and  sciences. 

21.  It  is  true,  that,  in  many  of  the  branches  which  have 
been  mentioned,  the  ordinary  business  is  frequently  transact- 
ed, and  the  mechanical  operations  performed,  by  persons  who 
have  not  been  regularly  instructed  in  a  course  of  mathemat- 
ics. Machines  are  framed,  lands  are  surveyed,  and  ships  are 
s4:eered,  by  men  who  have  never  thoroughly  investigated  the 
principles,  which  lie  at  the  foundation  of  their  respective 
arts.  The  reason  of  this  is,  that  the  methods  of  proceeding, 
in  their  several  occupations,  have  been  pointed  out  to  them, 
by  the  genius  and  labour  of  others.  The  mechanic  often 
works  by  rules,  which  men  of  science  have  provided  for  his 
use,  and  of  which  he  knows  nothing  more,  than  the  practical 
application.  The  mariner  calculates  his  longitude  by  tables, 
for  which  he  is  indebted  to  mathematicians  and  astronomers 
of  no  ordinary  attainments.  In  this  manner,  even  the  ab- 
struse parts  of  the  mathematics  are  made  to  contribute  their 
aid  to  the  common  arts  of  life. 

22.  But  an  additional  and  more  important  advantage,  to 
persons  of  a  liberal  education,  is  to  be  found,  in  the  enlarge- 
ment and  improvement  of  the  reasoning  powers.  The  mind, 
like  the  body,  acquires  strength  by  exertion.  The  art  of 
i»easoning,  like  other  arts,  is  learned  by  practice.  It  is  per- 
fected, only  by  long  continued  exercise.  Mathematical  stu- 
dies are  peculiarly  fitted  for  this  discipline  of  the  mind.  They 
are  calculated  to  form  it  to  habits  of  fixed  attention  ;  of  sa- 
gacity, in  detecting  sophistry  ;  of  caution,  in  the  admission  of 
proof ;  of  dexterity,  in  the  arrangement  of  arguments  •,  and 
of  skill,  in  making  all  the  parts  of  a  long  continued  process 
tend  to  a  result,  in  which  the  truth  is  clearly  and  firmly  es- 
tabhslied.  When  a  habit  of  close  and  accurate  thinking  is 
tlius  acquired  ;  it  may  be  applied  to  any  subject,  on  which  a 
man  of  letters  or  of  business  may  be  called  to  employ  his 
talents.  "  The  youth,"  says  Plato,  "  who  are  furnished  with 
nia^ematical  knowledge,  are  prompt  and  quick,  at  all  other 
:-'.:iences." 

It  is  not  pretended,  that  an  attentioo  to  other  objects  of 
inquiry,  is  rendered  unnecessary,  by  the  study  of  the  mathe- 
matics. It  is  not  their  office,  to  lay  before  us  historical  facts  ; 
•/>  teach  the  principles  of  morals:  to  store  the  fancy  wilh 


MATHEMATICS.  7 

brilliant  images ;  or  to  enable  us  to  speak  and  write  with 
•rhetorical  vigour  and  elegance.  The  beneficial  effects  which 
they  produce  on  the  mind,  are  to  be  seen,  principally,  in  the 
regulation  and  increased  energy  of  the  reasoning  pozvers. 
These  they  are  calculated  to  call  into  frequent  and  vigorous 
exercise.  At  the  same  time,  mathematical  studies  may  be  so 
conducted,  as  not  often  to  require  excessive  exertion  and  fa- 
tigue. Beginning  with  the  more  simple  subjects,  and  ascend- 
ing gradually  to  those  which  are  more  complicated  ^  the  mind 
acquires  strength,  as  it  advances;  and  by  a  succession  of 
steps,  rising  regularly  one  above  another,  is  enabled  to  sur- 
mount the  obstacles  which  lie  in  its  way.  Tn  a  course  of 
mathematics,  the  parts  succeed  each  other  in  such  a  connect- 
ed series,  that  the  preceding  propositions  are  preparatory  to 
those  which  follow.  The  student  who  has  made  himself 
master  of  the  former,  is  qualified  for  a  successful  investiga- 
tion of  the  latter.  But  he  who  has  passed  over  any  of  the 
ground  superficially,  will  find  that  the  obstructions  to  his  fu- 
ture progress  are  yet  to  be  removed.  In  mathematics,  as  m 
war,  it  should  be  made  a  principle,  not  to  advance,  while  any 
thing  is  left  unconquered  behind.  It  is  important  that  the 
student  should  be  deeply  impressed  with  a  conviction  of  the 
necessity  of  this.  Neither  is  it  sufliicient  that  he  understands 
the  nature  of  one  proposition  or  method  of  operation,  before 
proceeding  to  another.  He  ought  also  to  make  himself /«- 
vniliar  with  every  step,  by  a  careful  attention  to  the  examples. 
He  must  not  expect  to  become  thoroughly  versed  in  the  sci- 
ence, by  merely  reading  the  main  principles,  rules  and  obser- 
vations. It  is  practice  only,  which  can  put  these  completely 
in  his  possession.  The  method  of  studying  here  recom- 
mended, is  not  only  that  which  promises  success,  but  that 
which  will  be  found,  in  the  end,  to  be  the  most  expeditious, 
and  by  far  the  most  pleasant.  While  a  superficial  attention 
occasions  perplexity  and  consequent  aversion ;  a  thorough 
investigation  is  rewarded  with  a  high  degree  of  gratification*. 
The  peculiar  entertainment  which  mathematical  studies  are 
calculated  to  furnish  to  the  mind,  is  reserved  for  those  who 
make  themselves  masters  of  the  subjects  to  which  their  at- 
tention is  called. 

Note.    The  principal  definitions,  theorems,  rules,  &c.  which  it  is  neces- 
sary to  commit  to  mtmory.  are  distinguished  by  being  put  in  Italics  or  Capi- 


ALGEBRA. 

SECTION  I. 
NOTATION,  NEGATIVE  QUANTITIES,  AXIOMS,  fyc 

Art.  23.  ALGEBRA  may  be  defined,  a  general  method 

OF  INVESTIGATING  THE  RELATIONS  OF  QUANTITIES,  BY  LET- 
TERS, AND  OTHER  SYMBOLS.  This,  it  Hiust  be  acknowledged, 
is  an  imperfect  account  of  the  subject ;  as  every  account 
must  necessarily  be,  which  is  comprised  in  the  compass  of  a 
definition.  Its  real  nature  is  to  be  learned,  rather  by  an  at- 
tentive  examination  of  its  parts,  than  from  any  summary  de- 
scription. 

The  solutions  in  Algebra,  are  of  a  more  general  nature, 
than  those  in  common  Arithmetic.     The  latter  relate  to  par- 
ticular numbers  ;  the  former,  to  whole  classes  of  quantities. 
On  this  account.  Algebra  has  been  termed  a  kind  of  universal 
Arithmetic.     The  generality  of  its  solutions  is  principally 
owing  to  the  use  of  letters^  instead  of  numeral  figures,  to 
express  the  several  quantities  which  are  subjected  to  calcu- 
lation.    In  Arithmetic,  when  a  problem  is  solved,  the  answer 
is  limited  to  the  particular  numbers  which  are  specified,  in 
the  statement  of  the  question.     But  an  algebraic  solution 
may  be  equally  applicable  to  all  other  quantities  which  have 
the  same  relations.     This  important  advantage  is  owing  to 
the  difference  between  the  customary  use  of  figures,  and  the 
manner  in  which  letters  are  employed  in  Algebra.     One  of 
the  nine  digits  invariably  expresses  the  same  number :  but  a 
letter  may  be  put  for  any  number  whatever.     The  figure  8 
always  signifies  eight ;  the  figure  5,  five,  &c.     And,  though 
one  of  the  digits,  in  connection  with  others,  may  have  a  local 
value,  different  from  its  simple  value  when  alone ;  yet  the 
same  combination  always  expresses  the  same  number.     Thus 
263  has  one  uniform  signification.     And  this  is  the  case  with 
every  other  combination  of  figures.     But  in  Algebra,  a  letter 
may  stand  for  any  quantity  which  we  wish  it  to  represent. 
Thus  h  may  be  put  for  2^  or  10,  or  50,  or  1000.     It  must  not 
be  understood  from  this,  however,  that  the   letter  has  no 


A-LGEBBA.  9 

determinate  value.  Ite  value  is  fixed  for  the  occasion.  For 
the  present  purpose,  it  remains  unaltered.  But  on  a  different 
occasion,  the  same  letter  may  be  put  for  any  other  number. 
A  calculation  may  be  greatly  abridged  by  the  use  of  let- 
fers  ;  especially  when  very  large  numbers  are  concerned. 
And  when  several  such  numbers  are  to  be  combined,  as  in 
multiplication,  the  process  becomes  extremely  tedious.  But 
a  single  letter  may  be  put  for  a  large  number,  as  well  as  for 
a  small  one.  The  numbers  26347297,  68347823,  and 
27462498,  for  instance,  may  be  expressed  by  the  letters  6,  c 
and  d.  The  multiplying  tliem  together,  as  will  be  seen 
hereafter,  will  be  nothing  more  than  writing  them,  one  after 
another,  in  the  form  of  a  word,  and  the  product  will  be  sim- 
ply bed.  Thus,  in  Algebra,  much  of  the  labour  of  calcula- 
tion may  be  saved,  by  the  rapidity  of  the  operations.  Solu- 
tions are  sometimes  effected,  in  the  compass  of  a  few  lines, 
w^hich,  in  commoa  Arithmetic,  must  be  extended  through 
many  pages. 

24.  Another  advantage  obtained  from  the  notation  by  let- 
ters instead  of  figures,  is,  that  the  several  quantities  which 
are  brought  into  calculation,  may  be  preserved  distinct  from 
each  other,  though  carried  through  a  number  of  complicated 
processes ;  whereas,  in  arithmetic,  they  are  so  blended  to'^ 
gether,  that  no  trace  is  left  of  what  they  were,  before  the 
operation  began. 

25.  Algebra  differs  farther  from  arithmetic,  in  making  use 
of  unknown  quantities,  in  carrying  on  its  operations.  In 
arithmetic,  all  the  quantities  which  enter  into  a  calculation 
must  be  knowft.  For  they  are  expressed  in  number?,.  And 
every  number  must  necessarily  be  a  determinate  quantity. 
But  in  algebra,  a  letter  may  be  put  for  a  quantity,  before  its 
value  has  been  ascertained.  And  yet  it  may  have  such  rela- 
tions to  other  quantities,  with  which  it  is  connected,  as  to 
answer  an  important  purpose  in  the  calculation. 

NOTATION. 

26.  To  facilitate  the  investigations  in  algebra,  the  several 
steps  of  the  reasoning,  instead  of  being  expressed  in  words^ 
are  translated  into  the  language  of  signs  and  symbols,  which 
may  be  considered  as  a  species  of  short-hand.  This  serves 
to  place  tlie  quantities  and  their  relations  distinctly  before 
the  eye,  and  to  bring  them  all  into  view  at  once.  They  are 
^thus  more  readily  compared  afid  understood,  thafi  when  re- 

3 


1.0  AEGEBRA. 

moved  at  a  distance  from  each  other,  as  in  the  common  mode 
of  writing.  But  before  any  one  can  avail  himself  of  this 
advantao;e,  he  must  become  perfectly  familiar  with  the  new 
language. 

27.  The  quantities  in  algebra,  as  has  been  already  observ- 
ed, arc  generally  expressed  by  letters.  The  first  letters  of 
the  alphabet  are  used,  to  represent  knozvn  quantities ;  and 
the  last  letters,  those  which  are  unknown.  Sometimes  the 
quantities,  instead  of  being  expressed  by  letters,  are  set  down 
in  figures,  as  in  common  arithmetic. 

28.  Besides  the  letters  and  figures,  there  are  certain  char- 
acters used,  to  indicate  the  relations  of  the  quantities,  or  the 
operations  which  are  performed  with  them.  Among  these 
are  the  signs  +  and  — ,  which  are  read  plus  and  minnsy  or 
more  and  less.  The  former  is  prefixed  to  quantities  which 
are  to  be  added ;  the  latter,  to  those  which  are  to  be  sub- 
tracted. Thus  a  +  b  signifies  that  b  is  to  be  added  to  a.  It 
is  read  a  plus  b,  or  a  added  to  b,  or  a  and  b.  If  the  expres- 
sion  be  a—b,  i.  e.  a  minus  by  it  indicates  that  b  is  to  be  sub- 
tracted from  a, 

29.  The  sign  -f  is  prefixed  to  quantities  which  are  con- 
sidered as  affii^mative  or  positive  ^  and  the  sign  — ,  to  those 
which  are  supposed  to  be  negative.  For  the  nature  of  this 
distinction,  see  art.  54. 

All  the  quantities  which  enter  into  an  algebraic  process, 
are  considered,  for  the  purposes  of  calculation,  as  either 
positive  or  negativ^e.  Before  the^r.^^  one,  unless  it  be  nega- 
tive, the  sign  is  generally  omitted.  But  it  is  always  to  be 
understood.     Thus  a  -f  6,  is  the  same  as  4-  c^  +  &• 

30.  Sometimes  both  -\-  and  —  are  prefixed  to  the  same 

letter.     The  sign  is  then  said  to  be  ambiguous.     Thus  a'^b 

signifies  that  in  certain  cases,  comprehended  in  a  general  so- 
lution, b  is  to  be  added  to  a,  and,  in  other  cases,  substracted 
from  it. 

31.  When  it  is  intended  to  express  the  difference  between 
two  quantities  without  deciding  which  is  the  one  to  be  sub 
tracted,  the  character  cr  or  -^  is  used.     Thus  a  -^  b,  or  a  ^  b 
denotes  the  difference  between  a  and  b,  without  determining 
whether  a  is  to  be  subtracted  from  b,  or  b  from  a, 

32.  The  equality  between  two  quantities  or  sets  of  quanti- 
ties is  expressed,  by  parallel  hues  =.  Thus  a  +  b  =  d  sig- 
nifies that  a  and  6  together  are  equal  to  d.     And  a  -f-  <?  =  n 


NOTATION.  1 1 

^  h  -\-  g  =i  h  signifies  that  a  and  d  equal  c,  which  is  equal 
to  b  and  g,  which  are  equal  to  A.     So  8  +  4  =  16  —  4  =  10 

+  2  =  7  +  2  +  3  =  12. 

33.  When  the  first  of  the  two  quantities  compared  is  great- 
er^ than  the  other,  the  character  >  is  placed  between  them. 
Thus  a  >  6  signifies  that  a  is  greater  than  b. 

If  the  first  is  less  than  the  other,  the  character  <  is  used ; 
as  a  <  6  ;  i.  e.  «  is  less  than  b.  In  both  cases,  the  quantity 
towards  which  the  character  opens,  is  greater  than  the  other. 

34.  A  numeral  figure  is  often  prefixed  to  a  letter.  This 
is  called  a  co-efficient.  It  shows  how  often  the  quantity  ex- 
pressed by  the  letter  is  to  be  taken.  Thus  26  signifies  twice 
b,  and  9b,  9  times  b,  or  9  multiplied  into  b. 

The  co-efficient  may  be  either  a  whole  number  or  a  frac- 
lion.  Thus  |6  is  two  thirds  of  b.  When  the  co-efficient  is 
not  expressed,  1  is  always  to  be  understood.  Thus  a  is  the 
same  as  1  a;  i.  e.  once  a, 

35.  The  co-efficient  may  be  a  letter,  as  well  as  a  figure. 
In  the  quantity  mb,  tu  may  be  considered  the  co-efficient  of 
h  ;  because  b  is  to  be  taken  as  many  times  as  there  are  units 
in  m.  If  m  stands  for  6,  then  mb  is  6  times  b.  In  3abc,  3 
may  be  considered  as  the  co-efficient  of  abc  ;  3a,  the  co-effi- 
cient of  be  ;  or  Sab,  the  co-efficient  of  c.     See  art.  42. 

36.  A  simple  quantity  is  either  a  single  letter  or  number, 
or  several  letters  connected  together,  without  the  signs  -h 
and  — .  Thus  a,  ab,  abd,  and  86  are  each  of  them  simple 
quantities.  A  compound  quantity  consists  of  a  number  of 
simple  quantities,  connected  by  the  sign  -f  or  — .  Thus 
<t  +  6,  d—y,  b—d-^  3h,  are  each  compound  quantities.  The 
members  of  which  it  is  composed,  are  called  terms, 

37.  If  there  are  two  terms  in  a  compound  quantity,  it  is 
called  a  binomiaL  Thus  «-{-6  and  a—b  are  binomials.  The 
latter  is  also  called  a  residual  quantity,  because  it  expresses 
the  difference  of  two  quantities,  or  the  remainder,  after  one 
IS  taken  from  the  other.  A  compound  quantity  consisting  of 
three  terms,  is  sometimes  called  a  trinomial  j  one  of  four 
terms,  a  quadrinomial,  Sic, 

38.  When  the  several  members  of  a  compound  quantity 
are  to  be  subjected  to  the  same  operation,  they  are  frequent- 
ly connected  by  a  line  called  a  vincidum.  Thus  a  -^b  -i-  c 
shows  that  the  sum  of  b  and  c  is  to  be  subtracted  from  a. 
But  «  — 6  +  c  'dignifies  that  b  only  is  to  be  subtracted  from  a, 


12  ALGEBRA. 

while  c  i«  to  be  added.  The  sum  of  c  and  c?,  subtracted  from 
the  sum  of  a  and  i,  is  «  +  6  —  c  -f  J,  The  marks  used  for 
parentheses,  (  )  are  often  substituted,  instead  of  a  hne,  for  a 
vincuhim.  Thug  a:— («  +  c)  is  the  same  as  as— «  +  c.  The 
equality  of  two  sets  of  quantities  is  expressed,  without  using 
a  vincuhim.  Thus  a  \- h  —  c  ■\-  d  signifies,  not  that  h  is 
equal  to  c  ;  but  that  the  sum  of  a  and  b  is  equal  to  the  sum 
of  c  and  cL 

39.  A  single  letter,  or  a  number  of  letters,  representing 
any  quantities  with  their  relations,  is  called  an  algebraic  ex- 
pression  ;  and  sometimes  a  formula*  Thus  «  +  i  +  3rf  is 
an  algebraic  expression. 

40.  The  character  X  denotes  multiplication.  Thus  a  xh 
is  a  multiplied  into  b  :  and  6  X  3  is  6  times  3,  or  6  into  3. 
Sometimes  a  point  is  used  to  indicate  multiplication.  Thus 
«.  J  is  the  same  as  a  x  b.  But  the  sign  of  multiplication  is 
more  commonly  omitted,  between  simple  quantities  ;  and  the 
letters  are  connected  together,  in  the  form  oi  a  word  or 
syllable.  Thus  ab  is  the  same  diS  a.b  or  a  x  b.  And  bcde 
is  the  same  as  5  X  c  X  c?  X  c.  When  a  compound  quantity 
is  to  be  multiplied,  a  vinculum  is  used,  as  in  the  case  of  sub- 
traction. Thus  the  sum  of  a  and  J,  multiplied  into  the  sum 
of  c  and  d,  is  a  -\-  b  x  c  +  c?,  or  («  4-  5)  X  (c  -f  d).  And 
(6  +  2)  X  5  is  8  X  5  or  40.  But  6  +  2  X  5  is  6  +  10  or 
16.  When  the  marks  of  parentheses  are  used,  the  sign  of 
multiplication  is  frequently  omitted.  Thus  (a;  +  y)  (cc  —  y) 
is  {x  -]ry)  X  {x—  y). 

41.  When  two  or  more  quantities  are  mHltiplied  together, 
each  of  them  is  called  a  factor*  In  the  product  ab^  a  is  a 
factor,  and  so  is  b.  In  the  product  x  X  a  •\-  m^  x  \^  one  of 
the  factors,  and  a  -\-  m^  the  other.  Hence  every  co-efficient 
may  be  considered  a  factor.  (Art.  35.)  In  the  product  Sy, 
3  is  a  factor,  as  well  as  y, 

42.  A  quantity  is  said  to  be  resolved  into  factors^  when  any 
fectors  are  taken,  which,  being  multiplied  together,  Avill  pro- 
duce the  given  quantity.  Thus  3ab  may  be  resolved  into 
the  two  factors  3«  and  &,  because  3a  x  b  is  Sab,  And 
bamn  may  be  resolved  into  the  three  factors  5  «,  and  m^  and 
n.  And  48  maybe  resolved,  into  the  two  factors  2  x  24,  or 
3  X  16,  or  4  X  12,  or  6x8;  or  into  the  three  factors 
^^  X  3  X  8,  OP  4  X  6  X  2,  &;c. 


NOTATION,  13 

43.  The  character  ~  is  ivsed  to  show,  that  the  quantity 
wliich  precedes  it,  is  k)  be  divided,  by  that  which  follows. 
Thus  a-r-  cisa  divided  by  c  ;  and  a  +  b-^c  +  dis  the  sum 
of  u  and  b,  divided  by  the  sum  of  c  and  d.  But  in  algebra, 
division  is  more  commonly  expressed,  by  writing  the  divisor 
under  the  dividend,  in  the  form  of  a  vulgar  fraction.  Thus 
a  c-^b  ^ 

T  is  the  sanxe  as  a  ~  6  :  and  TTT  ^^  ^^^  difference  of  c  and  i 

divided  by  the  sum  of  d  and  A.  A  character  prefixed  to  the 
dividing  line  of  a  fractional  expression,  is  to  be  understood 
as  referring  to  all  the  parts  taken  collectively ;  that  is,  to 

the  whole  value  of  the  quotient.     Thus  q,  —  — rj-  signifies 

that  the  quotient  of  6  +  c  divided  by  m  +  ^  is  to  be  sul> 

c  —  d       h-\-  n 

tracted  from  a.     And  — ; —  X denotes  that  the  first 

a  +m       X  —  y 

quotient  is  to  be  multiplied  into  the  second. 

44.  When  four  quantities  are  proportional,  the  proportion 
is  expressed  by  points,  in  the  same  manner,  as  in  the  Rule  of 
Three  in  arithmetic.  Thus  a:b::c  :d  signifies  that  a  has  to 
&,  the  same  ratio,  which  c  has  to  d.  And  ab  :  c  d::a  +  m; 
b  +  n,  means,  that  «  6  is  to  c  c? ;  as  the  sum  of  a  and  m,  to 
the  sum  of  b  and  n, 

45.  Algebraic  quantities  are  said  to  be  alike,  when  they 
are  expressed  by  the  same  letters,  and  are  of  the  s^me power  : 
and  unlike,  when  the  letters  are  different,  or  when  the  same 
letter  is  raised  to  different  powers.*  Thus  ab,  Sab,  --ah, 
and  —6ab,  are  like  quantities,  because  the  letters  are  the 
same  in  each,  although  the  signs  and  co-efficients  are  differ- 
ent. But  3a,  3y,  and  ob x,  are  unlike  quantities,  because 
the  letters  are  unlike,  although  there  is  no  difference  in  the 
signs  and  co-efficients. 

46.  One  quantity  is  said  to  be  a  multiple  of  another,  when 
the  former  contains  the  latter  a  certain  number  of  times 
without  a  remainder.  Thus  10^  is  a  multiple  of  2«;  and 
24  is  a  multiple  of  6. 

47.  One  quantity  is  said  to  be  a  measure  of  another,  when 
the  former  is  contained  in  the  latter,  any  number  of  times, 
without  a  remainder.  Thus  36  is  a  measure  of  1 5h  :  and  7 
is  a  measure  of  35. 

*  For  the  notation  of  powers  and  roots,  see  tJie  sections  on  thoige  subjects. 


H 


ALGE^KA. 


48.  The  value  of  an  expression,  is  the  nuniber  or  quantitj> 
for  which  the  expression  stands.  Thus  the  value  of  3  -f  4 
is  7;  of  3  X  4is  12;  of  V  is  2. 

49.  TJie  RECIPROCAL  of  a  quantity^  is  the  quotient  arising 

from  dividing  a  unit  by  that  quantity.     Thus  the  reciprocal 

1  .        1  . 

of  a  is  - ;  the  reciprocal  of  «  -f  6  is  ""TTT;  the  reciprocal 

of  4  is  -. 

50.  The  relations  of  quantities,  which,  in  ordinary  lan- 
guage, are  signified  by  words,  are  represented,  in  the  alge- 
braic notation,  by  signs*  The  latter  mode  of  expressing 
these  relations,  ought  to  be  made  so  famihar  to  the  mathe- 
matical student,  that  he  can,  at  any  time,  substitute  the  one 
for  the  other.  A  few  examples  are  here  added,  in  which, 
words  are  to  be  converted  into  signs. 

1  *  What  is  the  algebraic  expression  for  the  following  state- 
ment, in  which,  the  letters  a,  b,  c,  &c.  may  be  supposed  to 
represent  any  given  quantities  ? 

The  product  of  «,  b,  and  c,  divided  by  the  difference  of  c 

and  €?,  is  equal  to  the  sum  of  b  and  c  added  to  15  times  h. 

ab  c 
Ans.    3  =  5  -f  c  4-  1 5  /i. 

2.  The  product  of  the  difference  of  a  and  h  into  the  sum 
of  b,  c,  and  J,  is  equal  to  37  times  m,  added  to  the  quotient 
of  b  divided  by  the  sum  of  h  and  b.     Ans. 

3.  The  sum  of  a  and  b,  is  to  the  quotient  of  b  divided  by 
f  ;  as  the  product  of  a  into  c,  to  1 2  times  A.     Ans. 

4.  The  sum  of  «,  6,  and  c  divided  by  six  times  their  pro- 
duct, is  equal  to  four  times  their  sum  diminished  by  d.     Ans* 

5.  The  quotient  of  6  divided  by  the  sum  of  a  and  5,  is 
equal  to  7  times  d,  diminished  by  the  quotient  of  &,  divided 
by  36.     Ans. 

51.  It  is  necessary  also,  to  be  able  to  reverse  what  is  done 
in  the  preceding  examples,  that  is,  to  translate  the  algebraic 
signs  into  common  language. 

What  will  the  following  expressions  become,  when  words 
are  substituted  for  the  signs  ? 
a-\-b  a 

Ans.  Tlie  sum  of  a  and  b  divided  by  h,  is  equal  to  the 
product  of  a,  6,  and  c  diminished  by  6  times  wi,  and  increased 
by  the  quotient  of  a  divided  by  the  sum  of  a  and  c. 


NOTATION. 

Sh-'C        ,      ; h 

2     Q{^  +  — -_ — :^^x  a'\'b  +  c  — 


13 


„       ,     c-^Qd  ;iSIT 

J~4         

4.  a  —  biaci: : 3  X  A  +  (?  +  y. 

«~^  d+ah^  j,a  X  J+1  _     cd 

52.  At  the  close  of  an  algebraic  process,  it  is  frequently 
necessary  to  restore  the  numbers,  for  which  letters  had  been 
substituted,  at  the  beginning.  In  doing  this,  the  sign  of  mul- 
tiplication must  not  be  omitted,  as  it  generally  is,  between 
factors,  expressed  by  letters.  Thus,  if  a  stands  for  3,  and  by, 
lor  four;  the  product  a  J  is  not  34,  but  3  X  4  i.  e.  12. 

In  the  following  example*, 

Let «  =  3  And  (?  =t  6. 

&  =  4  m  =  8. 

e  =  2  w=10. 


Then, 

a 
1.  - 

4-  rn 
cd 

b 
+  - 

c  —  n 
3d 

3  +  8 
~2X6  + 

4  X  2-  10 
3X6 

1^ 

b  +  ad 

c  —  dm 

—  b  cmn 

-  4C7l 

yab      "" 

3. 

bm 

c^-f 

ab  — 
cd 

3d 
m 

3b 
4rt 

n  —  bc       b 
+  ^cd"^  a 

;=  

53.  An  algebrait:  expression,  in  which  numbers  have  been 
substituted  for  letters,  may  often  be  rendered  much  more 
simple,  by  reducing  several  terms  to  one.  This  can  not 
generally  be  done,  while  the  letters  remain.  If  a  ■\-  b  is  used 
for  the  sum  of  two  quantities,  a  can  not  be  united  in  the  same 
term  with  b.  But  if  a  stands  for  3,  and  b,  for  4,  then  a  -{-b 
=  34-4  =  7.  The  value  of  an  expression  consisting  of 
many  terms  may  thus  be  found,  by  actually  performing,  with 
the  numbers,  the  operations  of  addition,  subtraction,  multi- 
plication. Sic,  indicated  by  the  algebraic  characters. 

Find  the  value  of  the  following  expressions,  in  which  the 
letters  are  supposed  to  stand  for  the  same  numbers,  as  in  the 
preceding  article. 


J6  ALGEBRA. 

ad  3X6 

26  2X4 

2.  alm-ir^^^  _  ^  +  2/^  =  3  X  4  X  8  +  ^j-ITg  +  2  XlO  = 


7W 


-   6 


3.  «  +  c  X  w  — •  m  + ;^  —  05  X    n  —  m  ^ 

7)1)   ""~    O 


4. \r  ahc 

n  —  a  n  —  be 


o. 


«c-i-5m                   J        Ad-hx  a-c 
^^-fT  "^'^^^""  ^^  +  n == 

POSITIVE  AND  NEGATIVE  QUANTITIES.*       , 

54.  To  one  who  has  just  entercid  on  the  study  of  algebra, 
there  is  generally  nothing  more  perplexing,  tlian  the  use  of 
what  are  called  negative  quantities.  He  supposes  he  is  about 
to  be  introduced  to  a  class  of  quantities,  which  are  entirely 
new  ;  a  sort  of  mathematical  nothings^  of  which  he  can  form 
no  distinct  conception.  As  positive  quantities  are  real,  he 
concludes  that  those  which  are  negative  must  be  imaginary^ 
But  this  is  owing  to  a  misapprehension  of  the  term  negative, 
as  used  in  the  mathematics. 

55.  A  NEGATIVE  QUANTITY  IS  ONE  WHICH    IS   REQUIRED   TO 

BE  SUBTRACTED.  When  several  quantities  enter  into  a  cal- 
culation, it  is  frequently  necessary  that  some  of  them  should 
be  adUed  together,  while  others  are  subtracted.  The  former 
are  called  affirmative  or  positive,  and  iire,  marked  with  the 
sign  +  ;  the  latter  are  termed  negative,  and  distinguished  by 
the  sign  — .  If,  for  instance,  the  profits  of  trade  are  the  sub- 
ject  of  calculation,  and  the  gain  is  considered  positive  ;  the 
loss  will  be  negative ;  because  the  latter  must  be  subtracted 
from  the  former,  to  determine  the  clear  profit.  If  the  sums 
of  a  book  account,  are  brought  into  an  algebraic  process,  the 
debt  and  the  credit  are  distinguished  by  opposite  signs.  If  a 
man  on  a  journey  is,  by  any  accident,  necessitated  to  return 
several  miles,  this  backward  motion  is  to  be  considered  nega- 
tive,  because  that,  in  determining  his  real  progress,  it  must 

*  On  the  subject  of  Negative  quantities,  see  Newton's  Universal  Arifh^ 
metic,  Maseres  on  the  Negative  Sign,  ftlansfield's  Mathematical  Essay?, 
and  Maclaurio's,  irinapson's,  Euler's,  Saunderson's  and  Liidlam's  Algebra.. 


NEGATIVBS..  t^T 

he  subtracted  from  the  distance  which  he  has  travelled  in 
the  opposite  direction.  If  the  ascent  of  a  bodj  from  the 
earth  be  called  positive,  its  descent  will  be  negative.  These, 
are  only  different  examples  of  the  same  general  principle. 
In  each  of  the  instances,  one  of  the  quantities  is  to  be  suh' 
traded  from  the  other. 

56.  The  terms  positive  and  negative,  as  used  in  the  mathe- 
matics, are  merely  relative.  They  imply  that  there  is,  ei- 
ther in  the  nature  of  the  quantities,  or  in  their  circum- 
stances, or  in  the  purposejs  which  they  are  to  answer  in 
calculation,  some  such  opposition  as  requires  that  one 
should  be  subtracted  from  the  other.  But  this  opposition  is 
not  that  of  existence  and  non-existence,  nor  of  one  thing 
greater  than  nothing,  and  another  less  than  nothing.  For,  in 
many  cases,  either  of  the  signs  may  be,  indifferently  and  at 
pleasure,  applied  to  the  very  same  quantity  ;  that  is,  the  two 
characters  may  change  places.  In  determining  the  progress 
of  a  ship,  for  instance,  her  easting  may  be  marked  -f ,  and  her 
westing  —  ;  or  the  westing  may  be  -4-,  and  the  easting  — . 
All  that  is  necessary  is,  that  the  two  signs  be  prefixed  to  the 
quantities,  in  such  a  manner  as  to  show,  which  are  to  be 
added,  and  which  subtracted.  In  different  processes,  they 
may  be  differently  applied^  On  one  occasion,  a  downward 
motion  may  be  called  positive,  and  on  another  occasion, 
negative. 

57.  In  every  algebraic  calculation,  some  one  of  the  quan- 
tities must  be  fixed  upon,  to  be  considered  positive.  All 
other  quantities  which  will  increase  this,  must  be  positive  al- 
so. But  those  which  will  tend  to  diminish  it,  must  be  neg- 
ative. ^  In  a  mercantile  concern,  if  ^he  stock  is  supposed  to 
he  positive,  the  profits  will  be  positive  ;  for  they  increase  the 
stock ;  they  are  to  be  added  to  it.  But  the  losses  will  be 
negative  ;  for  they  diminish  the  stock  ;  they  are  to  be  sub- 
tracted from  it.  Whei^  a  boat,  in  attempting  to  ascend  a 
river,  is  occasionally  driven  back  by  the  current ;  if  the 
progress  up  the  stream,  to  any  particular  point,  is  considered 
positive,  every  succeeding  instance  of  forward  motion  wiH 
be  positive,  while  the  backward  motion  will  be  negative. 

58.  A  negative  quantity  is  frequently  gy^eater^  than  the 
positive  one  with  which  it  is  connected.  But  how,  it  may 
be  asked,  can  tlie  former  be  subtracted  from  the  latter  '/ 
The  greater  is  certainly  not  contained  in  the  less  :  how  then 
can  it  be  taken  out  of  'it  ?    The  jin^wer  to  \\m  is\  that  Ihc 

4 


J  5  ALGEBRA. 

greater  may  be  supposed  first  to  exhaust  the  less,  and  thei> 
to  leave  a  remainder  equal  to  the  difference  between  the 
two.  If  a  man  has  in  his  possession,  1000  dollars,  and  has 
contracted  a  debt  of  1 500 ;  the  latter  subtracted  from  the 
foj-mer,  not  only  exhausts  the  whole  of  it,  but  leaves  a  bal- 
ance of  500  against  him.  In  common  language,  he  is  500 
dollars  worse  than  nothing. 

59.  In  this  way,  it  frequently  happens,  in  the  course  of  an 
algebraic  process,  that  a  negative  quantity  is  brought  to  stand 
alone.  It  has  the  sign  of  subtraction,  without  being  connec- 
ted with  any  other  quantity,  from  which  it  is  to  be  subtrac- 
ted. This  denotes  that  a  previous  subtraction  has  left  a  re*^ 
mainder,  which  is  a  part  of  the  quantity  subtracted.  If  the 
latitude  of  a  ship  which  is  20  degrees  north  of  the  equator, 
is  considered  positive,  and  if  she  sails  south  25  degrees  j 
her  motion  first  diminishes  her  latitude,  then  reduces  it  to 
nothing,  and  finally  gives  her  5  degrees  of  south  latitude. 
The  sign  —  prefixed  to  the  25  degrees,  is  retained  before 
the  5,  to  show  that  this  is  what  remains  of  the  southzoard  mo- 
tion, after  balancing  the  20  degrees  of  north  latitude.  If 
the  motion  southward  is  only  fifteen  degrees,  the  remainder 
must  be  -|-  5,  instead  of  —  5,  to  show  that  it  is  a  part  of  the 
ship's  northern  latitude,  which  has  been  thus  far  diminished, 
but  not  reduced  to  nothing.  The  balance  of  a  book  account 
will  be  positive  or  negative,  according  as  the  debt  or  the 
credit  is  the  greater  of  the  two.  To  determine  to  which 
side  the  remainder  belongs,  the  sign  mnst  be  retained,  though 
there  is  no  other  quantity,  from  which  this  is  again  to  be  sub- 
tracted, or  to  which  it  is  to  be  added. 

60.  When  a  quantity  continually  decreasing  is  reddfced  to 
nothing,  it  is  sometimes  said  to  become  afterwards  less  than 
nothing.  But  this  is  an  exceptionable  manner  of  speaking.* 
No  quantity  can  be  really  less  than  nothing.  It  may  be  di- 
minished, till  it  vanishes,  and  gives  place  to  an  opposite  quan- 
tity. The  latitude  of  a  ship  crossing  the  equator,  is  first 
made  less,  then  nothing,  and  afterwards  contrary  to  what  it 
was  before.  The  north  and  south  latitudes  may  therefore 
be  properly  distinguished,  by  the  signs  -h   and  —  ;  all  the 

*  Note.  The  expression  "  less  than  nothing,'*'^  may  not  be  wholly  im- 
proper ;  if  it  is  intended  to  be  understood,  not  literally,  but  merely  as  a  con- 
venient phrase  adopted  for  the  sake  of  avoiding  a  tedious  circumlocution  ; 
as  we  say  "the  sun  rises,"  instead  of  saying  "the  earth  rolls  round,  and 
brings  the  sun  into  view."  The  use  of  it  in  this  manner,  is  warranted  by 
jNewlon,  Euler,  and  others. 


NEGATIVES.  1& 

positive  degrees  being  on  one  side  of  0,  and  all  the  negative, 
on  the  other  ;  thus, 

+  6, +  5, +  4,  -f  3,+  l,0,  ~i,-2,  -  3,  -  4,-  5,  &c. 
The  numbers  belonging  to  any  other  series  of  opposite 
quantities,  may  be  arranged  in  a  similar  manner.  So  that  0 
may  be  conceived  to  be  a  kind  of  dixiding  point  between 
positive  and  negative  numbers.  On  a  thermometer,  the  de- 
grees aho-ce  0  may  be  considered  positive,  and  those  helow  0, 
negative. 

61.  A  quantity  i&  sometimes  said  to  be  subtracted  from  0. 
By  this  is  meant,  that  it  belongs  on  the  negative  side  of  0. 
But  a  quantity  is  said  to  be  added  to  0,  when  it  belongs  on 
the  positive  side.  Thus,  in  speaking  of  the  degrees  of  a 
thermometer,  0+6  means  6  degrees  above  0 ;  and  0  ~  6,  6 

-degrees  below  0. 

AXIOMS. 

62.  The  object  of  mathematical  inquiry  is,  generally,  to 
mvestigate  some  unknown  quantity,  and  discover  how  great 
it  is.  This  is  effected,  by  comparing  it  with  some  other 
quantity  or  quantities  already  known,  The  dimensions  of 
a  stick  of  timber  are  found,  by  applying  to  it  a  measuring 
rule  of  known  length.  The  weight  of  a  body  is  ascertained, 
by  placing  it  in  one  scale  of  a  balance,  and  observing  how 
many  pounds  in  the  opposite  scale,  will  equal  it.  And  any 
quantity  is  determined,  when  it  is  found  to  be  equal  to 
some  known  quantity  or  quantities. 

Let  a  and  b  be  known  quantities,  and  ?/,  one  which  is  un- 
known. Then  y  will  become  known,  if  it  is  discovered  to 
be  equal  to  the  sum  of  a  and  b  ;  that  is,  if 

y  ■=:  a-^  b. 
An  expression  like  this,  representing  the  equality  between 
one  quantity  or  set  of  quantities,  and  another,  is  called  an 
equation.  It  will  be  seen  hereafter,  that  much  of  the  busi- 
ness of  algebra  consists  in  finding  equations,  in  which  some 
unknown  quantity  is  shown  to  be  equal  to  others  which  are 
known.  But  it  is  not  often  the  fact,  that  the  first  compari- 
son of  the  quantities,  furnishes  the  equation  required.  It 
will  generally  be  necessary  to  make  a  number  of  additions, 
subtractions,  multiplications,  &c.  before  the  unknown  quan- 
tity is  discovered.  But  in  all  these  changes,  a  constant 
equality  must  be  preserved,  between  the  two  sets  of  quanti- 
tities  compared.     This  will  be  done,  if,  in  making  the  altera- 


20  '      ALGEBRA. 

tions,  we  are  guided  by  the  following  axioms.  These  are 
not  inserted  here,  for  the  purpose  of  being  proved  ;  for  they 
are  self-evident.  (Art.  10.)  But  as  they  must  be  continu- 
ally introduced  or  implied,  in  demonstrations  and  the  solu- 
tions of  problems,  they  are  placed  together,  for  the  conve- 
nience of  reference. 

63.  y\xiom  1.  If  the  same  quantity  or  equal  quantities 
be  added  to  equal  quantities,  their  sums  will  be  equal. 

2.  If  the  same  quantity  or  equal  quantities  be  subtracted 
from  equal  quantities,  the  remainders  will  be  equal. 

3.  If  equal  quantities  be  multiplied  into  the  same,  or  equal 
quantities,  the  products  will  be  equal. 

4.  If  equal  quantities  be  divided  by  the  same  or  equal 
quantities,  the  quotients  will  be  equal. 

5.  If  the  same  quantity  be  both  added  to  and  subtracted 
from  another,  the  value  of  the  latter  will  not  be  altered. 

6.  If  a  quantity  be  both  multiplied  and  divided  by  another, 
the  value  of  the  former  will  not  be  altered. 

7.  If  to  unequal  quantities,  equals  be  added,  the  greater 
will  give  the  greater  sum. 

8.  If  from  unequal  quantities,  equals  be  subtracted,  the 
gi'eater  will  give  the  greater  remainder. 

9.  If  unequal  quantities  be  multiplied  by  equals,  the  great- 
er will  give  the  greater  product. 

10.  If  unequal  quantities  be  divided  by  equals,  the  great- 
er will  give  the  greater  quotient. 

1 1 .  Quantities  which  are  respectively  equal  to  any  other 
quantity,  are  equal  to  each  other. 

12.  The  whole  of  a  quantity  is  greater  than  a  part. 
This  is,  by  no  means,  a  complete  list  of  the  self-evident 

propositions,  which  are  furnished  by  the  mathematics.  It  is 
not  necessary  to  enumerate  them  all.  Those  have  been  se- 
lected, to  which  we  shall  have  the  most  frequent  occasion  to 
refer. 

64.  The  investigations  in  algebra  are  carried  on  principal- 
ly, by  means  of  a  series  of  equations  and  proportions.  But 
instead  of  entering  directly  upon  these,  it  will  be  necessary 
to  attend,  in  the  first  place,  to  a  number  of  processes,  on 
which  the  management  of  equations  and  proportions  de- 
pends. These  preparatory  operations  are  similar  to  the  cal- 
culations under  the  common  rules  of  arithmetic.  We  have 
addition,  multiplication,  division,  involution,  &c.  in  algebra, 
as  well  as  in  arithmetic.     But  this  application  of  a  common 


ADDITION.  21 

name,  to  operations  in  these  two  branches  of  the  mathemat- 
ics, is  often  the  occasion  of  perplexity  and  mistake.  The 
learner  naturally  expects  to  find  addition  in  algebra  the  same 
as  addition  in  arithmetic.  They  are  in  fact  the  same,  in 
many  respects  :  in  all  respects  perhaps,  in  which  the  steps  of 
the  one  will  admit  of  a  direct  comparison,  with  those  of  the 
other.  But  addition  in  algebra  is  more  extensive^  than  in 
arithmetic.  The  same  observation  may  be  made,  concern- 
ing several  other  operations  in  algebra.  They  are,  in  many 
points  of  view,  the  same  as  those  which  bear  the  same  names 
in  arithmetic.  But  they  are  frequently  extended  farther, 
and  comprehend  processes  which  are  unknown  to  arithme- 
tic. This  is  commonly  owing  to  the  introduction  of  nega- 
tive quantities.  The  management  of  these  requires  step§ 
w,hich  are  unnecessary,  where  quantities  of  one  class  only 
are  concerned.  It  will  be  important  therefore,  as  we  pass 
along,  to  mark  the  difference^  as  well  as  the  resemblance^  be- 
tween arithmetic  and  algebra ;  and,  in  some  instances,  to 
give  a  new  definition,  accommodated  to  the  latter. 


■<E^9^i>- 


SECTION  II. 


ADDITION. 


Art.  Q6.  xN  entering  on  an  algebraic  calculation,  the  first 
4'hing  to  be  done,  is  evidently  to  collect  the  materials.  Sev- 
eral distinct  quantities  are  to  be  concerned  in  the  process. 
These  must  be  brought  together.  They  must  be  connected 
in  some  form  of  expression,  which  will  present  them  at  once 
i':^  our  view,  and  show  the  relations  which  they  have  to  each 
other.  This  collecting  of  quantities  is  what,  in  algebra,  is 
called  ADDITION.     It  may  be  defined,  the  connecting  of 

SEVERAL   quantities,    WITH    THEIR  SIGNS,  IN  ONE  ALGEBRAIC 
EXPRESSION. 


22  ALGEBRA. 

66.  It  is  common  to  include  in  ihe  defmition,  "  uniting  in 
one  term,  such  quantities,  as  will  admit  of  being  united.'^ 
But  this  is  not  so  much  a  part  of  the  addition  itself,  as  a 
reduction^  which  accompanies  or  follows  it.  The  addition 
may,  in  all  cases,  be  performed,  by  merely  connecting  the 
quantities,  by  their  proper  signs.  Thus  a  added  to  b  is,  evi- 
dently, a  and  b  r  that  is,  according  to  tl>e  algebraic  notation 
a-^b*  And  a,  added  to  the  sum  of  b  and  c,  is  a-\-b-\-c. 
And  a-^-b,  added  to  c^d,  is  a-H6+c-fJ.  In  the  same  man- 
ner, if  the  sum  of  any  quantities  whatever,  be  added  to  the 
sum  of  any  others,  the  expression  for  the  whole,  will  contain 
all  these  quantities  connected  by  the  sign  -J-. 

67.  Again,  if  the  difference  of  a  and  b  be  added  to  c ;  the 
sum  will  he  a—b  added  to  c,  that  is,  «— &+c.  And  if  «— & 
be  added  to  c—d,  the  sum  will  be  a—b-^-c—d.  In  one  of 
the  compound  quantities  added  here,  a  is  to  be  diminished 
2>y  h^  and  in  the  other,  c  is  to  be  diminished  by  d ;  the  sunt 
of  a  and  c  must  therefore  be  diminished,  both  by  6,  and  by 
d,  that  is,  the  expression  for  the  sum  total,  must  contain 
—b  and  —d»  On  the  same  principle,  all  the  quantities 
which,  in  the  parts  to  be  added,  have  the  negative  sign,  must 
retain  this  sign,  in  the  amount.  Thus  a+26-~c,  added  to 
d—k—m,  is  a-\-2b  —  c-\-d—h—m, 

68.  The  sign  must  be  retained  also,  when  a  positive  quan- 
tity is  to  be  added,  to  a  single  negative  quantity.  If  a  be  ad- 
ded to  —by  the  sum  will  be  —  5-f «.  Here  it  may  be  objec- 
ted, that  the  negative  sign  prefixed  to  &,  shows  that  it  is  to  be 
subtracted^  What  propriety  then  can  there  be  in  adding  it  ? 
In  reply  to  this,  it  may  be  observed,  than  the  sign  prefixed 
to  b  while  standing  alone,  signifies  that  b  is  to  be  subtracted, 
not  from  or,  but  from  some  other  quantity,  which  is  not  here 
expressed.  Thus  —b  may  represent  the  loss^  which  is  to  be 
subtracted  from  the  stock  in  trade.  (Art.  bo,)  The  object 
of  the  calculation,  however,  may  not  require  that  the  value 
of  this  stock  should  be  specified.  But  the  loss  is  to  be  con- 
nected with  a  profit  on  some  other  article.  Suppose  the  prof- 
it is  2000  dollars,  and  the  loss  400,  The  inquiry  then  is, 
what  is  the  value  of  2000  dollars  profit,  when  connected  with 
400  dollars  loss  ? 

The  answer  is,  evidently,  2000  —  400,  which  shows  that 
SOOO  dollars  are  to  be  added  to  the  stock,  and  400  subtracted 
from  it ;  or,  which  will  amount  to  the  same,  that  the  differ- 
ence between  2000  and  400  is  to  be  added  to  the  stock. 


ADDITION< 


m 


69.  Quantities  are  added,  then,  by  writing  them  one 

AFTER  another,  WITHOUT  ALTERING  THEIR  SIGNS  ;    observing 

always,  that  a  quantity,  to  which  no  sign  is  prefixed,  is  to  he 
considered  positive.     (Art.  29.) 

The  sum  of  a-{-m,  and  Z>  — 8,  and  '2h--3m-\-d,  and  k—ni 
and  r-\-3m—7/,  is 

a-{m+b'-S-i-'2h—3m+d-\-h—n-{'r+3m—y. 

70.  It  is  immaterial  in  what  order  the  terms  are  arranged. 
The  sum  of  a  and  h  and  c  is  either  a-{'b  +  c,  or  <x-\-c-\-h,  or 
c+6  +  «»  For  it  evidently  makes  no  difference,  which  of 
the  quantities  is  added  Jirst.  The  sum  of  6  and  3  and  9,  is 
the  same  as  3  and  9  and  6,  or  9  and  6  and  3. 

And  a+m—n,  is  the  same  as  a.-'n+m.  For  it  is  plainly 
of  no  consequence,  whether  we  first  add  m  to  a,  and  after- 
wards subtract  n  ;  or  first  subtract  n,  and  then  add  m. 

71.  Though  connecting  quantities  by  their  signs,  is  all 
which  is  essential  to  addition  ;  yet  it  is  desirable  to  make  the 
expression  as  simple  as  may  be,  hy  reducing  several  terms  to 
one.     The  amount  of  3a,  and  6J,  and  4a,  and  5&,  is 

3a-i-6Z»4-4a4-56. 
But  this  may  be  abridged.  The  first  and  third  terms  may 
be  brought  into  one  ;  and  so  may  the  second  and  fourth* 
For  3  times  a,  and  4  times  a,  make  7  times  a.  And  6  times 
/>,  and  5  times  Z»,  make  1 1  times  b.  The  sum,  when  redu- 
ced, is  therefore  la-\-\\h. 

For  making  the  reductions  coimected  with  addition,  two 
rules  are  given,  adapted  to  the  two  cases,  in  one  of  which,, 
the  quantities  and  signs  are  alike,  and  in  the  other,  the  quan- 
tities are  alike,  but  the  signs  are  unlike.  Like  quantities  are- 
the  same  pozvers  of  the  same  letters.  (Art.  45.)  But  as  the 
addition  of  powers  and  radical  quantities  will  be  considered, 
in  a  future  section,  the  examples  given  in  this  place,  will  be. 
all  of  the  first  power. 

72.  Case  I.     To  reduce  several  terms  to  one,  when 

THE  QUANTITIES  ARE  ALIKE  AND  THE  SIGNS  ALIKE,  ADD  THE 
CO-EFFICIENTS,  ANNEX  THE  COMMON  LETTER  OR  LETTERS,  AND 
PREFIX  THE  COMMON  SIGN. 

Thus,  to  reduce  35-f  76,  that  is -|- 36 -f  76  to  one  term, 
add  the  co-efiicients  3  and  7,  to  the  sum  10,  annex  the  com- 
mon letter  6,  and  prefix  the  sign  -\- .  The  expression  will 
then  be  4-106.  That  3  times  any  quantity,  and  7  times  the 
same  quantity,  make  1 0  times  that  quantity,  needs  no  proof. 


24 


1 

ALGEBRA. 

Examples, 

ho 
%c 
9bc 
3bc 

3xy 

Ixy 

xy 

2xy 

7b  +  xy           ry'\'3abk 
nb-^Sxi/         Sry-\-  abh 
%-^2xy         6ry'\'4abh 
6b -{-5X7/         2ry-\-  abh 

cdxy  +  3mg 
2cdxy-\'  mg 
5cdxy-\-7mg 
Icdxy  +  Smg 

15bc  23b-\-llxy  I5cdxy+ldmg 


The  mode  of  proceeding  will  be  the  same,  if  the  signs 
are  negative. 
Thus  —3bc^bc  —  5bc,  becomes,  when  reduced,  —  9&c. 
And '—ax^3ax—2ax=i  —  6ax.     Or  thus, 


-3bc 

—  ax 

—2ab^  my 

—  3ach—8bdy 

-  be 

^3ax 

—  ab—Smy 

—  ach—-  bdy 

-5bc 

— 2flx 

—7ab  —  Smy 

'-bach—7bdy 

dbc  ^10ab  —  l2my 


73.  It  may  perhaps  be  asked  here,  as  in  art.  6S,  what  pro- 
priety there  is,  in  adding  quantities,  to  which  the  negative 
sign  is  prefixed  ;  a  sign  which  denotes  subtraction  ?  The  an- 
swer to  this  is,  that  when  the  negative  sign  is  applied  to  seve- 
ral quantities,  it  is  intended  to  indicate  that  these  quantities 
are  to  be  subtracted,  not  from  each  other  ^  but  from  some  oth- 
er quantity,  marked  with  the  contrary  sign.  Suppose  that, 
in  estimating  a  man's  property,  the  sum  of  money  in  his  pos- 
session is  marked  +,  and  the  debts  which  he  owes  are  mark- 
ed — .  If  these  debts  are  200,  300,  500  and  700  dollars,  and 
if  a  is  put  for  100  ;  they  will  together  be  —2a— 3a— 5a— 7«. 
And  the  several  terms  reduced  to  one,  will  evidently  be  —  1 7a5 
that  is,  1700  dollars. 

74.  Case  IL  To  reduce  several  terms  to  one,  when 
the  quantities  are  alike,  but  the  signs  unlike,  take  the 
less  co-efficient  from  the  greater  ;  to  the  difference, 
annex  the  common  letter  or  letters,  and  prefix  the 
sign  of  the  greater  co-efficient. 

Thus,  instead  of  8a— 6a,  we  may  write  2a. 
And  instead  of  7&  — 26,  we  may  put  6b, 
For  the  simple  expression,  in  each  of  these  instances,  is 
equivalent  to  the  compound  one,  for  which  it  is  substituted* 


ADDlfigN.  05 

To     +eb       +4h         5hc       ^hm     -  d[y+ 6w       3h—  dx     ,       .^ 
Add   -4b       -Gb     -Ibc    -Ohm         4dy-  m       5h-\-4dx    iiiSJ 

Sam -f  26  —26c  My-\-5m  ^IH3Bi^ 

75.  Here  again,  it  may  excite  surprise,  that  what  appears 
to  be  subtraction,  should  be  introduced  under  addition.  But 
according  to  what  has  been  observed,  (Art.  66,)  this  subtrac- 
tion is,  strictly  speaking,  no  part  of  the  addition.  It  belongs" 
to  a  consequent  reduction.  Suppose  66  is  to  be  added  t© 
a— 46.     The  sum  is  a— 464-66.     (Art.  69.) 

But  this  expression  may  be  rendered  more  simple.  As  it 
now  stands,  46  is  to  be  subtracted  from  a,  and  66  added. 
But  the  amount  will  be  the  same,  if,  without  subtracting  any 
thing,  we  add  26,  making  the  whole  «  +  26.  And  in  all  sim- 
ilar instances,  the  balance  of  two  or  more  quantities,  may  be 
substituted  for  the  quantities  themselves. 

77.  If  two  equal  quantities  have  contrary  signs,  they  de- 
stroy each  other,  and  may  be^cancelled.     Thus  +66—66* 

=0:     Aod  3x^.- 18=0:  And  76c - 76c =0. 

Let  there  be  any  two  quantities  whatever,  of  which  a  is 
the  greater,  and  6  the  less. 

Their  sum  will  be       a-{-b 

And  their  difference  a—b 


The  sum  and  difference  added,  will  be  2«-f  0,  or  simpiy 
2a.  That  is,  if  the  sum  and  difference  of  any  two  quantities 
be  added  together,  the  whole  will  be  twice  the  greater  quan- 
tity. This  is  one  instance,  among  multitudes,  of  the  rapidity 
with  which  general  truths  are  discovered  and  demonstrated 
in  algebra.     (Art.  23.) 

78.  If  several  positive,  and  several  negative  quantities  are 
to  be  reduced  to  one  term  5  first  reduce  those  which  are 
positive,  next  those  which  are  negative,  and  then  take  the 
difference  of  the  co-efficients,  of  the  two  terms  thus  found. 


Ex.  1.  Reduce  136+66-f  6-46-56-76,  to  one  term, 

6=     206  ) 
76= -166  5 


By  art.  72,  136 +  66 -f   6=     206 
And  —46-56 


ue 


By  art.   74,  206  —  1 66 =46,  which  is  tTie  vol- 

of  all  the  givqn  quantities,  taken  together. 


26  ALGEBRA. 

Ex.2.  Reduce  Sxy— a;y+2a;i/— 7a;^-i-4x^— 9a;?/4-7jc^«-6«y, 
The  positive  terms  are  Sxy      The  negative  terms  are  —  xy 

2x2/  —  Ixy 

Axy  —  ^xy 

Ixy  —  6a??/ 

And  their  sum  is       16a:^  —  23a^«/ 

Then  16xy'-23xy=  —  7xy. 

Ex.  3.    3acZ— 6ac?+aJ4-7«(?— 2ac?+9arf— 8acZ— 4«^=:0. 

4.  2abm—ahm+7abm—3ahm+7abm  = 

5,  axy—laxy+Saxy—axy—Saxy-f-Oaxy^ 

79.  If  the  letters,  in  the  several  terms  to  he  added,  are 
different,  they  can  only  be  placed  after  each  other,  witli 
their  proper  signs.  They  cannot  be  miited  in  one  simple 
term.  If  46,  and  —  6y,  and  3x,  and  17/i,  and  —5c/,  and^, 
be  added  ;  their  sum  will  be 

46 -62/+ 3a: +17^ -56?+ 6.  (Art  69.) 
Different  letters  can  no  more  be  united  in  the  same  term, 
than  dollars  and  guineas  can  be  added,  so  as  to  make  a  sin- 
gle sum.  Six  guineas  and  4  dollars  are  neither  ten  guineas 
nor  ten  dollars.  Seven  hundred,  and  five  dozen,  are  neither 
12  hundred  nor  12  dozen.  But,  in  such  cases,  the  algebraic 
signs  serve  to  show  how  the  different  quantities  stand  related 
to  each  other  ;  and  to  indicate  future  operations,  which  are 
to  be  performed,  whenever  the  letters  are  converted  into 
numbers.  In  the  expression  a +6,  the  two  terms  can  not  be 
united  in  one.  But  if  a  stands  for  15,  and  if,  in  the  course 
of  a  calculation,  this  number  is  restored;  then  «+6  w^ill 
become  15  +  6,  which  is  equivalent  to  the  single  term  21. 
In  the  same  manner,  a— 6  becomes  15  —  6,  which  is  equal  to 
9.  The  signs  keep  in  view  the  relations  of  the  quantities, 
till  an  opportunity  occurs  of  reducing  several  terms  to  one. 

80.  When  the  quantities  to  be  added  contain  several 
•terms  which  are  alike,  and  several  which  are  unlike,  it  will 
be  convenient  to  arrange  them  in  such  a  manner,  that  the 
similar  terms  may  stand  one  under  another. 

To         Sbc—6d-{-2h—Sy  ')    These  may  be  arranged  thus. 

Add  Sbc+xSd-^bg     S      Sbc-Qd+2b—Sy 

And       2d-^y-{-2x+b        )  -Sbc-Sd  +  x+hg 

2d         +  y+Sx        +h 

The  sum  will  be  -7c?+26-2i/-h4aH'%+^- 


SUBTRACTION.  27 


Examples, 

i.  Add  and  reduce  a6-h8  to  C(Z— 3  and  ^a6— 4m-h2. 
The  sum  is6a^+7+c£?— 4?w. 

2.  Add  x-\-^y^dx^  to  7— j:— 8-fAm. 
Ans.  3y  —  rfa:  —  1  +  ^?«' 

3.  Add  abm—Sx+hm,  to  y— :r4-7,  and  5^—6^+9. 

4.  Add  3am4-6--7a:^— 8,  to  10a:y— 9 -f  5am. 

5.  Add  Qahy+ld—l+mxy,  to  3a%— 7<?-|-17~m:ri/* 

6.  Add  lad^h+Qxy-^adj  to  5ad+h—lxy. 

7.  Add  3a6— 2ay+a?,  to  db—ay-\'hx^h» 

8.  Add  262^— 3ax+2a,  to  3&a:—%+a. 


■«^9^^' 


SECTION  IH. 


subtraction- 


Art.  81.  Addition  is  bringing  quantities  together,  to 
tind  their  amount.  On  the  contrary,  Subtraction  is  find- 
ing THE  DIFFERENCE  OF  TWO  QUANTITIES,  OR  SETS  OF  QUAN- 
TITIES. 

Particular  rules  might  be  given,  for  the  several  cases  in 
subtraction.  But  it  is  more  convenient  to  have  one  general 
rule,  founded  on  the  principle,  that  taking  away  SL^^positive 
quantity,  from  an  algebraic  expression,  is  the  same  in  effect, 
as  «nnea:mg  an  equal /legafiue  quantity  ;  and  taking  away  a 
negative  quantity  is  the  same,  as  annexing  an  equal  positive 
one. 

Suppose  +&  is  to  be  subtracted  from  a-^-b. 

Taking  away  -{-  5,  from  «  4-  6,  leaves  «. 

And  annexing  —  6,  to  «4-^,  gives  «+^-"^* 

But  by  axiom  5th,  a+b-^b  h  equal  to  a 


28  ALGEBRA. 

That  is,  taking  atoay  a  positive  term,  from  an  algebraic  ex- 
pression, is  the  same  in  effect,  as  annexing  an  equal  negative 
term. 

Again,  suppose  —b  is  to  be  subtracted  from         a—h 
Taking  away  —6,  from  a— 6,  leaves  a 

And  annexing  +6,  to  a—b,  gives  a—b'\-b 

But  a—b-{-b  is  equal  to  a 

That  is,  taking  away  a  negative  term,  is  equivalent  to  an- 
nexing  a  positive  one.     If  an  estate  is  encumbered  with  a 
debt ;  to  cancel  this  debt,  is  to  add  so  much  to  the  value  of 
the  estate.     Subtracting  an  item  from  one  side  of  a  book  ac- 
count, will  produce  the  same  alteration  in  the  balance,  as 
adding  an  equal  sum  to  the  opposite  side. 
To  place  this  in  another  point  of  view. 
If  m  is  added  to  &,  the  sum  is  by  the  notation,        b-\-m'> 
But  if  m  is  subtracted  from  5,  the  remainder  is     b—m  J 
So  if  m  and  h  are  each  added  to  5,  the  sum  is       &+m-{-/*i 
But  if  m  and  h  are  each  subtracted  from  6,  the  re-  > 

mainder  is  6— m  — ^) 

The  only  difference  then  between  adding  a  positive  quan- 
tity and  subtracting  it,  is,  that  the  sign  is  changed  from  -f 
to  —. 

Again,  if  m— n  is  subtracted  from  6,  the  remainder  is, 
b—m-\-n. 
For  the  less  the  quantity  subtracted,  the  greater  will  be  the 
remainder.  But  in  the  expression  m—n,m  is  diminished  by 
n  ;  therefore,  b—m  must  be  increased  by  n ;  so  as  to  become 
h—m-\-n  :  that  is,  m—n  is  subtracted  from  b,  by  changing 
+m  into  — w,  and  — n  into  -fn,  and  then  writing  them  after 
h,  as  in  addition.  The  'explanation  will  be  the  same,  if 
there  are  several  quantities  which  have  the  negative  sign. 
Hence, 

82.  To  PERFORM  SUBTRACTION  IN  ALGEBRA,  CHANGE  THE 
SIGNS  OF  ALL  THE  QUANTITIES  TO  BE  SUBTRACTED,  OR  SUP- 
POSE THEM  TO  BE  CHANGED,  FROM  +  TO  — ,  OR  FROM  —  TO  -f-, 
AND  TH^N  PROCEED  AS  IN   ADDITION. 

The  signs  are  to  be  changed,  in  the  subtrahend  only. 
Those  in  the  minuend  are  not  to  be  altered.  Although  the 
rule  here  given  is  adapted  to  every  case  of  subtraction  ;  yet 
there  may  be  an  advantage  in  giving  some  of  the  examples 
in  distinct  classes. 

83.  In  the  first  place,  the  signs  may  be  alike,  and  the  min- 
.^lend  greater  than  the  subtrahend. 


SUBTRACTION.  29 

From  +28       166       14da       -28       -166       -lAda 

Subtract      +16       126         6da       -16       -126       -  eda 


Difference  +12  46  Sda  -12  -46  -8da 
Here,  in  the  first  example,  the  +  before  16  is  supposed 
to  be  changed  into  — ,  and  then,  the  signs  being  unlike,  the 
two  terms  are  brought  into  one,  by  the  second  case  of  re- 
duction in  addition.  (Art.  74.)  The  two  next  examples  are 
subtracted  in  the  same  way.  In  the  three  last,  the  —  in  the 
subtrahend,  is  supposed  to  be  changed  into  + .  It  may  be 
well  for  the  learner,  at  first,  to  write  out  the  examples  ;  and 
actually  to  change  the  signs,  instead  of  merely  conceiving 
them  to  be  changed.  When  he  has  become  familiar  witJb 
the  operation,  he  can  save  himself  the  trouble  of  transcrib- 
ing. 

This  case  is  the  same  as  subtraction  in  arithmetic.     The 
two  next  cases  do  not  occur  in  common  arithmetic. 

84.  In  the  second  place,  the  signs  may  be  ahke,  and  the 
minuend  less  than  the  subtrahend. 

From  +166       126  6da         -16         -126       -  eda 

Sub.     +286       166         14da         -28         -166       -14da 


Dif.      -126     -46       -Sda         +12  46  8da 

The  same  quantities  are  given  here,  as  in  the  preceding 
article,  for  the  purpose  of  comparing  them  together.  But 
the  minuend  and  subtrahend  are  made  to  change  places. 
The  mode  of  subtracting  is  the  same.  In  this  class,  a  greater 
quantity  is  taken  from  a  less :  in  the  preceding,  a  less  from  a 
greater.  By  comparing  them,  it  will  be  seen,  that  there  is 
no  difference  in  the  answers,  except  that  the  signs  are  oppo- 
site.  Thus  166  —  126  is  the  same  as  126  —  166,  except  that 
oile  is  +  46,  and  the  other  —  46  :  That  is,  a  greater  quantity 
subtracted  from  a  less,  gives  the  same  result,  as  a  less  sub- 
tracted from  a  greater,  except  that  the  one  is  positive,  and 
the  other  negative.    See  art.  58  and  59. 

85.  In  the  third  place,  the  signs  may  be  unlike. 

From  +28       +166       +14da       -28       -166       -14da 
Sub.     -16       -126       -   6da       +16       +126       +   Gda 


Dif.      +44  286  '20da      -44       -286        -20c/a 


30  ALGEBRA. 

From  these  examples,  it  will  be  seen  that  the  difference 
between  a  positive  and  -a  negative  quantity,  may  be  greater 
than  either  of  the  two  quantities.  In  a  thermometer,  the 
clifference  between  28  degrees  above  cypher,  and  16  below, 
is  44  degrees.  The  difference  between  gaining  1000  dol- 
lars in  trade,  and  losing  500,  is  equivalent  to  1 500  dollars. 

^6»  Subtraction  may  be  proved,  as  in  arithmetic,  by  ad- 
ding the  remainder  to  the  subtrahend.  The  sum  ought  to 
be  equal  to  the  minuend,  upon  the  obvious  principle,  that 
the  difference  of  two  quantities  added  to  one  of  them,  is  e- 
qual  to  the  other.  This  serves  not  only  to  correct  any  par- 
ticular errour,  but  to  verify  the  general  rule. 

From      2*2/— 1  h'\-3bx  hy-^  ah  nd^lhy 

Sub.      — a?y-f7         2h^9bx        5hy—6ah        5nd^  by 

Dif.        SxyS  -Uy-^-Sah 

From     3abm^  xy     —  174-4aa;  a^-f-  76         Sah+axy 

Sob*    —lahm'\'Qxy     —20—  ax     — 4aa;-f  1^     —lah+axy 

Rem.  lOabm—lxy  Sax—  Sb 

87»  When  there  are  several  terms  alike,  they  may  be  re- 
duced as  in  addition. 

1.  From  «6,  subtract  3am -|- am +7«7w-i- 2am -f-6«?n. 

Ans.  aft  — 3am— am— 7am— 2am— 6am=a6— 19am.(Art.72.) 

2.  From  y,  subtract  —a-— a— a— a. 
Ans.  ?/ -i- a -f  a-f  a -h a = 2/ -f  4a. 

'    3.  From  ax^bc-^Sax+lbc,  subtract  4ftc— 2aa?-f  6c-f  4aoC. 
Ans.  a:e— 6c-l-3aa7-f  *75c— 4&c+2aa:— 5c— 4aa::^2aa?-f  5c. 
(Art.  78.) 

4.  From  ad-{-Sdc—bx,  subtract  3ad-{-lbx—dc'\'ad. 

88.  When  the  letters  in  the  minuend  are  different  from 
those  in  the  subtrahend,  the  latter  are  subtracted,  by  first 
changing  the  signs,  and  then  placing  the  several  terms  one 
after  another,  as  in  addition.  (Art.  79.) 

From  Sab  -{-8—my+dh,  subtract  x—dr+ 4>hy  —  bmx, 
Au5.  daIf-{-S--my-\-dh-'X-{'dr—^hy-{-hmx^ 


MULTIPLICATION.  -i 

88.b.  The  sign  — ,  placed  before  the  marks  of  parenthesis 
which  include  a  number  of  quantities,  requires,  that  when 
these  marks  are  removed,  the  signs  of  all  the  quantities  thus- 
included,  should  be  changed. 

Thus  a—  (b—c-hd)  signifies  that  the  quantities  5,  — c,  and 
+  d,  are  to  be  subtracted  from  a.  The  expression  will  then 
become  a  —  b+C'-d. 

2.  I3ad-\'xy-\'d—  (lad^3py-{-d+hm'-ry)  =6a(;?4-2T^— 
hm-^-ry. 

3.  labc'-S+'7x—{Sahc—S''dx'{'r)^Aabc-{-lx-{'dX'-u 

4.  3«df4-^— 2^^— (Ty-f  3A— wa?+4aff— %—a6?)  = 

5.  6awj— %+8  —  (l6-f3Jy— S+aw— e4-r)  = 

6.  7a3^~22?-h5~(4+^~ai/+a:+36)  = 

88,c.  On  the  other  hand,  when  a  number  of  quantities  are 
introduced  within  the  marks  of  parenthesis,  with  —  immedi- 
ately preceding  ;  the  signs  must  be  changed. 

Thus  --m  +  6-«fx+3A  =  — (?»-64•£?ic~3/^) 


CC^^^' 


SECTION  IV. 


MULTIPLICATION.* 

Art.  89.  JLn  addition,  one  quantity  is  connected  witl. 
another.    It  is  frequently  the  case,  that  the  quantities  brought 
together  are  equal  ;  that  is,  a  quantity  is  added  to  itself. 
As  a-l-a=2a  a-4-a-fa-|-a=4a 

This  repeated  addition  of  a  quantity  to  itself,  is  what  was 
originally  called  multiplication.     But  the  term,  as  it  is  now 

*  Newton^a  UnivevsEj  Arithmetic,  p.  4.  Maseres  on  the  Negative  SigOy 
Sec.  II.  Camus'  Arithmetic,  Book  II.  Chap.  3.  Euler's  Algebra,  Sec.  I. 
and  II.  Chap  3.  Simpson's  Algebra,  8ec.  IV.  Maclaurin,  Saunderson, 
Laccoix,  Ludlam. 


32  ALGEBRA. 

used,  has  a  more  extensive  signification.  We  have  frequent 
occasion  to  repeat,  not  only  the  whole  of  a  quantity,  but  a 
certain  portion  of  it.  If  the  stock  of  an  incorporated  com- 
pany is  divided  into  shares,  one  man  may  own  ten  of  them, 
another  five,  and  another  a  part  only  of  a  share,  say  two 
fifths.  When  a  dividend  is  made,  of  a  certain  sum  on  a 
share,  the  first  is  entitled  to  ten  times  this  sum,  the  second  to 
Jive  times,  and  the  third  to  only  two  fifths  of  it.  As  the  ap- 
portioning of  the  dividend,  in  each  of  these  instances,  is  up- 
on the  same  principle,  it  is  called  multiplication  in  the  last, 
as  well  as  in  the  two  first. 

Again,  suppose  a  man  is  obligated  to  pay  an  annuity  of  100 
dollars  a  year.  As  this  is  to  be  subtracted  from  his  estate,  it 
may  be  represented  by  — «.  As  it  is  to  be  subtracted  year 
after year^  it  will  become,  in  four  years,  —a—a^a—a^.—  4a, 
This  repeated  subtraction  is  also  called  multiplication.  Ac- 
cording to  this  view  of  the  subject  ; 

90.  Multiplying  by  a  whole  number  is  taking  the 
multiplicand  as  many  times,  as  there  are  units  in  the 
multiplier. 

Multiplying  by  1 ,  is  taking  the  multiplicand  once^  as  a. 
Multiplying  by  2,  is  taking  the  multiplicand  t-wice^  as  a-\-a. 
Multiplying  by  3,  is  taking  the  multiplicand  three   times,  as 

Multiplying  by  a  fraction  is  taking  a  certain  por- 
tion OF  THE  MULTIPLICAND  AS  MANY  TIMES,  AS  THERJS  ARE 
LIKE  PORTIONS  OF  AN  UNIT  IN  THE  MULTIPLIER.* 

Multiplying  by  },  is  taking  ]  of  the  multiplicand,  once,  as  \  «. 
Multiplying  by  |,  is  taking  }  of  the  multiplicand,  twice,  as 

}«  +  }«. 
Multiplying  by  |,  is  taking  i  of  the  multiplicand  three  times. 

Hence,  if  the  multiplier  is  a  unit,  the  product  is  equal  to^ 
the  multiplicand  :  If  the  multiplier  is  greater  than  a  unit, 
the  product  is  greater  than  the  multiplicand :     And  if  the 
multiplier  is  less  than  a  unit,  the  product  is  less  than  the 
multiplicand. 

Multiplication  by  a  negative  quantity,  has  the  same 
relation    to   multiplication  by   a   positive   quantity, 

WHICH     SUBTRACTION     HAS    TO    ADDITION.       In    thc    OUC,    the 

sum  of  the  repetitions  of  the  multiplicand  is  to  be  addedy 
to  the  other  quantities  with  which  this  multiplier  is  connec- 
ted.    In  the  other,  the  sum  of  these  repetitions  is  to  be  sub- 

*  Se?  Note  C. 


MULTIPLICATION.  3j 

traded  from  the  other  quantities.  This  subtraction  is  per- 
formed at  the  time  of  multiplying,  by  changing  the  sign  of 
the  product.     See  Art.  107  and  108. 

91.  Every  multipher  is  to  be  considered  a  mmher.  We 
sometimes  speak  of  multiplying  by  a  given  weighty  or  meas- 
ure^  a  sum  of  money ^  (^c.  But  this  is  abbreviated  language. 
If  construed  literally,  it  is  absurd.  Multiplying  is  taking 
either  the  whole  or  a  part  of  a  quantity,  a  certain  number  ojf 
times.  To  say  that  one  quantity  is  repeated  as  many  times, 
as  another  is  heavy,  is-  nonsense.  But  if  a  part  of  the  weight 
of  a  body  be  fixed  upon  as  a  unit,  a  quantity  may  be  multr- 
plied  by  a  number  equal  to  the  number  of  theseparts  con- 
tained in  the  body.  If  a  diamond  is  sold  by  weight,  a  par- 
ticular price  may  be  agreed  upon  for  each  grain.  A  grain  is 
here  the  unit ;  and  it  is  evident  that  the  value  of  the  dia- 
mond, is  equal  to  the  given  price  repeated  as  many  times, 
as  there  are  grains  in  the  whole  weight.  We  say  concisely, 
that  the  price  is  multiplied  by  the  weight ;  meaning  that  it 
is  multiplied  by  a  number  equal  to  the  number  of  grains  in 
the  weight.  In  a  similar  manner,  any  quantity  whatever 
may  be  supposed  to  be  made  up  of  parts,  each  being  consid- 
ered a  unit,  and  any  number  of  these  may  become  a  multi- 
plier. 

92.  As  multiplying  is  taking  the  whole  or  a  part  of  a 
quantity  a  certain  number  of  times,  it  is  evident  that  the 
■product  must  be  of  the  same  nature  as  the  multiplicand. 

If  the  multiplicand  is^an  abstract  number^  the  product 
will  be  a  number,  .  . 

If  the  multiplicand  is  weight,  the  product  will  be  weight. 
If  the  multiplicand  is  a  line,  the  product  will  be  a  line.  Re- 
peating a  quantity  does  not  alter  its  nature.  It  is  frequently 
said,  that  the  product  of  two  lines  is  a  surface,  and  that  the 
product  of  three  lines  is  a  solid.  But  these  are  abbreviated 
expressions,  which  if  interpreted  literally  are  not  correct. 
See  Section  xxi. 

93.  The  multiplication  o^  fractions  will  be  the  subject  of 
a  future  section.  We  have  first  to  attend  to  multiplication 
by  positive  whole  numbers.  This,  according  to  the  defini- 
tion (Art.  90.)  is  taking  the  multiplicand  as  many  times,  as 
there  are  units  in  the  multipher.  Suppose  a  is  to  be  multi- 
plied by  b,  and  'that  b  stands  for  3.  There  are,  then,  three 
imits  in  the  multiplier  b.  The  multiplicand  must  therefore 
be  taken  three  times :  thas,  «4-ff4-«^3a.  w  la* 


31  ALGEBRA. 

So  that,  multiplying  two  letters  together  is  nothing  mdi% 
than  writing  them  one  after  the  other ^  either  with,  or  without 
the  sign  of  multiphcation  between  them.  Thus  h  mutipU- 
ed  into  c,  is  6  X  c,  or  he.     And  x  into  y,  is  a?  x  y^  or  x,y,  or  xy. 

94,  If  more  than  two  letters  are  to  be  multiphed,  they 
must  be  connected  in  the  same  manner.  Thus  a  into  b  and 
c,  is  cba.  For  bj  the  last  article,  a  into  6,  is  ha.  This  pro- 
duct is  noAv  to  be  multiplied  into  c.  If  c  stands  for  5,  then 
ha  is  to  be  taken  five  times,  thus, 

ba'\-ha-\-ha+ba-^ba=5ba,  ov  cba. 
The  same  explanation  may  be  applied  to  any  number   of 
letters.     Thus  am  into  ccy,  is  amxy.     And  bh  into  mrx,  is 
hhmrx. 

95.  It  is  immaterial  in  what  order  the  letters  are  arranged. 
The  product  ha  is  the  same  as  ah.  Three  times  five  is  equal 
to  five  times  three.  Let  the  number  5  be  represented  by  as 
many  points,  in  a  horizontal  line  ;  and  the  number  3,  by  as 
many  points  in  a  perpendicular  line. 


Here  it  is  evident  that  the  whole  number  of  points  is  equal, 
either  to  the  number  in  the  horizontal  row  three  times  repeat- 
ed, or  to  the  number  in  the  perpendicular  row  five  times  re- 
peated ;  that  is,  to  5  X  3,  or  3x5.  This  explanation  may.be 
extended  to  a  series  of  factors  consisting  of  any  numbers 
whatever.  For  the  product  of  two  of  the  factors  may  be 
considered  as  one  number.  This  may  be  placed  before  or 
after  a-  third  factor :  the  product  of  three,  before  or  after  a 
fourth,  (Szc. 

Thus  24  =  4x6  or  6X4  =  4X3X2  or  4X2X3  or  2X3X4. 

The  product  of  a,  b,  c,  and  d^  is  ahcd,  or  acdbj  or  dcha,  or  hade. 

It  will  generally  be  convenient,  however,  to  place  the  letters 
in  alphabetical  order. 

96.  When  the  letters  have  numerical  co-efficients, 
these  must  be  multiplied  together,  and  prefixed  to 
the  product  of  the  letters. 

Thus  Sa  into  2h  is  6ab.  For  if  a  into  h  is  ab,  then  3  times 
a  into  5,  is  evidently  3ah  :  and  if,  instead  of  multiplying  by 
&,  we  multiply  by  tivice  b,  the  product  must  be  twice  as  great, 
that  is  2Xo«3  or  Gab. 


MULTIPLICATION.  3^^ 

Mult.         9ah         12hy        Sdk  2ad  Ibdh        Say 

Into  Sxy  2rx  my         IShmg  x  Smx 


Prod.       21ahxy  Sdhmy  Ibdhx 

97.  If  either  of  the  factors  consist  of  figures  only,  these 
must  he  multipUed  into  the  co-efficients  and  letters  of  the 
other  factors. 

Thus  Sab  into  4,  is  I2ab.  And  36  into  2a;,  is  72a?.  And 
24  into  %5  is  24%. 

98.  If  the  multiplicand  is  a  compound  quantity,  each  of  its 
terms  must  be  multiplied  into  the  multiplier.  Thus  b  +  c+d 
into  a  is  ab  •{■  ac -i- ad.  For  the  whole  of  the  multiplicand  is 
to  be  taken  as  many  times,  as  there  are  units  in  the  multi- 
pHer.  If  then  a  stands  for  3,  the  repetitions  of  the  multipli- 
cand are 

b  +  c-{-d 
b  +  c-]-d 
b-{-c-{-d 


And  their  sum  is       36  +  30+ 3c?,  that  is,  aJ+rtc+^c/. 

Mult.     d-\-2xi/  2h-{-m  3hl+l  2hm-\-3  +  dr 

Into      36  Gdi/  my  46 


Prod.    3bd-\-6bxy  3hlmi/-\-my 


99.  The  preceding  instances  must  not  be  confounded 
with  those  in  which  several  factors  are  connected  by  the 
sign  X  5  or  by  a  point.  In  the  latter  case,  the  multipHer  is 
to  be  written  before  the  other  factors  without  being  repeated. 
The  product  of  6  x  c?  into  «,  is  06  x  d,  and  not  ab  x  ad.  For 
hxdis  bd,  and  this  into  a,  is  abd,  (Art.  94.)  The  expression 
bxd  is  not  to  be  considered,  hke  6  + J,  a  compound  quantity 
consisting  of  two  terms.  Different  terms  are  always  separa- 
ted by  +  or  — .  (Art.  36.)  The  product  of  bxhxmxy 
into  a,  is  axbxhxmXy  or  abhmy.  But  6  +  A+7n+y  into 
a,  is  ab-{-ah-jram+ay, 

100.  If  both  the  factors  are  compound  quantities,  each 
term  in  the  multiplier  must  be  multiplied  into  each  in  (he  muU 
tiplicand. 

Thus    a+6  into  c-^d  is  ac+ad+bc-^-bd. 


36  ALGEBRA. 

For  the  units  in  the  multiplier  a+b  are  equal  to  the  units 
in  a  added  to  the  units  in  b.     Therefore  the  product  produ- 
ced hy  a,  must  be  added  to  the  product  produced  by  b. 
The  product  of  c+d  into  a  is  ac-\-ad   )     .        ^ko 
The  product  of  c+d  into  b  is  bc+bd   5    ^^^*  ^^' 
The  product  of  c+^  into  a-\-b  is  therefore  ac+ad-\-bc-\-bd. 
Mult.  3x+d  4ay+2b       a+1 

Into      2a-\-hm  3c  +rx     3x  +  4 


Prod.  6aa;4-2a(?-f  3Amx-|-<?Am  3ax+Sx+4a-\-4, 


Mult.  2^+7  into  Gd+1.     Prod.  12J/i+42(!+2/i4  7. 
Mult,  diz-hrx  +  h  into  6m-f4  +  7y.     Prod. 
Mult.    7  +  6i+aJinto  3r+4+2A.     Prod. 

101.  When  several  terms  in  the  product  are  alike,  it  will 
be  expedient  to  set  one  under  the  other,  and  then  to  unite 
them,  by  the  rules  for  the  reduction  in  addition. 

Mult.  &+«  &+C+2     •  «+  3/+1 

Into      b+a  bJrc+3  3b-\-2x-\-l 


bb^ab  bb'\-bc+2b 

-f«i+aa  be         +cc4-2c 

+  36  -f-3c4-6 


Prod,  bb 4- 2ab  +  aa        bb  +  ^bc  +  db  +  cc  +  dc-^e 


Mult.  3fl+r^+4  into  2a-f3J+l.  Prod. 
Mult.  &+cJ+2  into  3&+4c<Z+7.  Prod. 
Mult.  3b'\'2x  +  h  into  axdx2x.     Prod. 

103.  It  will  be  easy  to  see  that  when  the  multipher  and 
multiplicand  consist  of  any  quantity  repeated  as  a  factor,  this 
factor  will  be  repeated  in  the  product,  as  many  times  as  in 
the  multiplier  and  multiphcand  together. 
Mult.  aXaXft  Here  a  is  repeated  three  times  as  a  factor- 
Into     ax  a  Here  it  is  repeated  twice. 

Prod,   axaxaxaxa.  Here  it  is  repeated  Jive  times. 

The  product  of  bbbb  into  bbb,  is  bbbbbbb. 
The  product  of  2a:  X  3x  X  4ac  into  5x  X  Qx,  is  2af  X  3a?  X  4a? 
X  5a?  X  Gx. 


MULTIPLICATION.  37 

104.  But  the  numeral  co-efficients  of  several  fellow-factors 
may  be  brought  together  by  multiplication.  f^j  'ij  ^ 

Thus  2a  X  36  into  4a  X  56  is  Set  X  36  x  4«  x  56,  or  i^CiaohL      r- j. 
For  the  co-efficients  zve  factors,  (Art.  41)  and  it  *sSii{ni(JjTpQ^^ 
terial  in  what  order  these  are  arranged.  (Art.  95.)     So  thST" 
2a  X  36  X  4a  X  56  «=2  X  3  X  4  X  5  X  a  X  a  X  6  xb  =  l^Oaahb. 
The  product  of  3a  x  46A  into  5m  X  Qy,  is  SGOabhmy. 
The  product  of  46x6ci  into  2a?-f  1,  is  ^1^8bdx+2^bd. 

105.  The  examples  in  multiplication  thus  far  have  been 
confined  to  positive  quantities.  It  will  now  be  necessary  to 
consider  in  what  manner  the  result  will  be  affected,  by  mul- 
tiplying positive  and  negative  quantities  together.  We  shall 
find, 

That     +  into  -\-  produces  + 

—  into  +  — 
-1-  into  —                   — 

—  into  —  + 

All  ^these  may  be  comprised  in  one  general  rule,  which  it 
will  be  important  to  have  always  famihar.     If  the  signs  op 

THE  FACTORS  ARE  ALIKE,  THE  SIGN  OF  THE  PRODUCT  WILL  BE 
AFFIRMATIVE  ;  BUT  IF  THE  SIGNS  OF  THE  FACTORS  ARE  UN- 
LIKE, THE  SIGN  OF  THE  PRODUCT  WILL  BE  NEGATIVE. 

106.  The  first  case,  that  of  -f  into  -1-,  needs  no  farthet 
illustration.  The  second  is  —  into  -f,  that  is,  the  multipH- 
cand  is  negative,  and  the  multiplier  positive.  Here  —a  in- 
to -|-4  is  —4a.     For  the  repetitions  of  the  multiplicand  are, 

— a— a— a— «  =  -— 4a. 

Mult.     6-3a  2a-m      A~3(f~4  a-2-7d-T 

Into     Gy  3h+x     2y  Sb-^-h 


Prod.  6by—lQay  2hy—edi/Sy 


107.  In  the  two  preceding  cases,  the  affirmative  sign  pre- 
fixed to  the  multiplier  shows,  that  the  repetitions  of  the  mul- 
tiplicand are  to  be  added,  to  the  other  quantities  with  which 
the  multiplier  is  connected.  But  in  the  two  remaining  cases, 
the  negative  sign  prefixed  to  the  multiplier,  indicates  that  the 
sum  of  the  repetitions  of  the  multiphcand  are  to  be  subtrac- 
ted from  the  other  quantities.  (Art.  90.)  And  this  subtrac- 
tion is  performed,  at  the  time  of  multiplying,  by  making  the 
i^ign  of  the  product  opposite  to  that  of  the  multiplicand. 


35  ALGEBRA. 

Thus  -f-a  into  —4,  is  —  4cr,    For  Ihc  repetitions  of  tlie  mul- 
tiplicand are, 

+  «  +  «+ /T+a=4-4a. 

But  this  sum  is  to  be  subtracted,  from  the  other  quantities 
with  whicn  the  multipher  is  connected.  It  will  then  become 
—4a.    (Art.  82.) 

Thus,  in  the  expression  h — (4  x  a),  it  is  manifest  that  4  X  « 
is  to  be  subtracted  from  b.  Now  4  x  a  is  4«,  that  is,  +  4a. 
But,  to  subtract  this  from  b,  the  sign  +  must  be  changed  into 
—.  So  thatfi  — (4xa)  is  b—4a.  And  ax  —4  is  there- 
fore —  4a. 

Again,  suppose  the  multiplicand  is  a,  and  the  multiplier 
(6  —  4).  As  (6  —  4)  is  equal  to  2,  the  product  will  be  equal 
to  2a.  This  is  less  than  the  product  of  6  into  a.  To  obtam 
then  the  product  of  the  compound  multiplier  (6—4)  into  a, 
we  must  subtract  the  product  of  the  negative  part,  from  that 
of  the  positive  part. 

JMultiplying        a )  .    .,  V  Multiplying  a 

Into  ^       ^  6-4 1  ''  t''«  ^^""^  ^'   I  Into  ^2 

And  the  product  6a— 4a,  is  the  same  as  the  product   2a. 
Therefore  a  into  —4,  is— 4a. 

But  if  the  multipher  had  been  (6  +  4),  the  two  products 
must  have  been  added. 

Multiplying        a )  .     ,  ^  Multiplying  a 

Into  6  +  45  '^^^^  ^™^  ^^  I  Into  1 0 

And  the  prod.     6a+4a  is  the  same  as  the  product     10a 

This  shows  at  once  the  difference  between  multiplying  by 
n  positive  factor,  and  multiplying  by  a  negative  one.  In  the 
former  case,  the  sum  of  the  repetitions  of  the  multiplicand 
is  to  be  added  to,  in  the  latter,  subtracted  froniy  the  other 
quantities,  with  which  the  multiplier  is  connected.  For  eve- 
ry negative  quantity  must  be  supposed  to  have  a  reference 
to  some  other  which  is  positive  ;  though  the  two  may  not  al- 
ways stand  in  connection,  when  the  multiplication  is  to  be 
performed. 

Mult,  a+b  3dy+hx-\-2         Sh  -f  3 

Into     b—x  mT'-ab  ad— 6 


Prod,  ab  +  bb-ax-bx  Sadh+Sad-lSh-lS 

108.  If  two  negatives  be  multiplied  together,  the  product 

will  be  affirmative  :— 4  x  —  «  =  +  4a.     In  this  case,  as  in  the 


MULTIPLICATION.  ^9 

preceding,  the  repetitions  of  the  multipHcand  are  to  be  sub" 
traded,  because  the  multipher  has  the  negative  sign.  These 
repetitions,  if  the  multipHcand  is  — «,  and  the  multipher  —4, 
are  —a  —a  — «--a=— 4a.  But  this  is  to  be  subtracted  by 
changing  the  sign.     It  then  becomes  +  4a. 

Suppose  —a  is  multiphed  into  (6  —  4).  As  6  — 4=2,  the 
product  is  evidently,  twice  the  multiplicand,  that  is  —2a. 
But  if  we  multiply  —a  into  6  and  4  separately ;  —a  into  6 
is  —Ga^  and  —a  into  4  is  —4a.  (Art.  106.)  As,  in  the  mul- 
tiplier, 4  is  to  be  subtracted  from  6  ;  so,  in  the  product,  —  4« 
must  be  subtracted  from  ^6a,  Now  —4a  becomes  by  sub- 
traction -f4a.  The  whole  product  then  is  —  6a+4a,  which 
is  equal  to  —2a.     Or  thus. 

Multiplying     -a  >  .     ,  \  Multiplying  -a 

Into    ^  6-45*^^^^  ^^^^  ^^  ( Into  2- 

And  the  prod.  —  6a -f  4a,  is  equal  to  tlie  product  — 2«. 
It  is  often  considered  a  great  mystery,  that  the  product  O'f 
two  negatives  should  be  affirmative.  But  it  amounts  to  no- 
thing more  than  this,  that  the  subtraction  of  a  negative  quan- 
tity, is  equivalent  to  the  addition  of  an  affirmative  one  ;  (Art, 
81,)  and,  therefore,  that  the  repeated  subtraction  of  a  nega- 
tive quantity,  is  equivalent  to  a  repeated  diddition  of  an  affirm- 
ative one.  Tal<:ing  off  from  a  man's  hands  a  debt  of  ten 
dollars  every  month,  is  adding  ten  dollars  a  month  to  the  val* 
ue  of  his  property. 

Mult,     a  — 4  3fZ— Ay— 2a?  3ai/— 5 

Into     Sb-G  46-7       -      6x-l 


Prod.  3a&  — 126  — 6a-}-24  ISaa;!/— 66aj— 3ay-f  6 

Multiply  Sad— ah  — 1  into  A—d^—hr, 
Multiply  2hy-{-3m—\  into  4(?— 2a; +3. 

109.  As  a  negative  multiplier  changes  the  sign  of  the 
quantity  which  it  multiplies ;  if  there  are  several  negative 
factors  to  be  multiplied  together. 

The  two  first  will  make  the  product  positive  ; 
The  third  will  make  it  negative; 
The  fourth  will  make  it  positive,  &;c. 
Thus  — ax— J  =  4-«&        ^  (   two  £diCior6 

-^abx —c^—abc       f   ,,  j     i.    r    )    three, 

-abcx-d==+abcd    I  the  product  of  j  yj,,„.; 


40  -     ALGEBRA. 

That  is,  the  product  of  any  even  number  of  negative  factors 
is  positive ;  but  the  product  of  any  odd  number  of  negative 
factors  is  negative. 

Thus  —a  X  —az=iaa  And  —a  X  —a  X  —a  x  —a^aaaa 

—ax  —ax  —a=::^aaa    —ax -ax -ax -ax  -a=i-aaaaa 

The  product  of  several  factors  which  are  all  positive,  is 
invariably  positive, 

110.  Positive  and  negative  terms  may  frequently  balance 
each  other,  so  as  to  disappear  in  the  product.  (Art.  77.)  A 
star  is  sometimes  put  in  the  place  of  the  deficient  term. 

Mult,  a  — &  mm—yy        aa-^-ah-^-hh  g 

Into     a-\-h  mm-\-yy  a—b 


aa—ab  aaa+aab-\-abb 

+ab—bb  —aab—abb—bbb 


Prod,  aa    ^  —bb  aaa    *        *     —bbb 


111.  For  many  purposes,  it  is  sufficient  merely  to  indicate 
the  multiplication  of  compound  quantities,  without  actually 
multiplying  the  several  terms.     Thus  the  product  of 

a+b  +  c  into  /t-f»i+y,  is  {a-^-b+c)  X  (A+wfr-f^)-  (Art  40.i 
The  product  of  ^ 

a-\-m  into  h-\-x  and  d+y,  is  {a+m)  X  {h+x)  X  {d-^-y). 
By  this  method  of  representing  multiplication,  an  important 
advantage  is  often  gained,  in  preserving  the  factors  distinct 
from  each  other. 

When  the  several  terms  are  multiplied  in  form,  the  ex- 
pression is  said  to  be  expanded.     Thus 

(a +6)  X  (c+d)  becomes  when  expanded  ac-\rcid-\-bc-\'bd, 

112.  With  a  given  multiplicand,  the  less  the  multiplier, 
the  less  will  be  the  product.  If  then  the  multiplier  be  redu- 
ced to  nothing,  the  product  will  be  nothing.  Thus  a  X  0=0. 
And  if  0  be  one  of  am/  number  of  fellow-factors,  the  product 
of  the  whole  will  be  nothing. 

Thus  abxcXSdxO  =  ^abcdxO=0. 
And  {a4-b)  x  (c+rf)  X  {h—m)  x  0=0. 

113.  Although,  for  the  sake  of  illustrating  the  different 
points  in  multiplication,  the  subject  has  been  drawn  out  into 
a  considerable  number  of  particulars ;  yet  it  will  scarcely  be 


DIVISION.  41 

necessary  for  the  learner,  after  he  has  become  familiar  with 
the  examples,  to  burden  his  memory  with  any  thing  more 
than  the  following  general  rule. 

Multiply  the  letters  and  co-epficients  of  each  term 
IN  the  multiplicand,  into  the  letters  and  co-efficients 

OF  EACH  TERM  IN  THE  MULTIPLIER  ;  AND  PREFIX,  TO  EACH 
TERM  OF  THE  PRODUCT,  THE  SIGN  REQUIRED  BY  THE  PRINCI- 
PLE, THAT  LIKE  SIGNS  PRODUCE   +,  AND  DIFFERENT  SIGNS  — . 

Mult.  «4"36— 2  into  4a-6i— 4. 
Mult.  4abxxx2  into  3my — 1  +  ^. 

Mult  {lah—y)x^  into  4a: X 3x5 X^.     . 

Mult.  (6a6-Ac?-f-l)  X  2  into  (8+4X-1)  xcf. 
Mult.  3«y-h3/— 44-A  into  (c?+^)  X  (A+y). 
Mult.  6aa?--(4^-J)  into  (6+l)x  (A+1). 
Mult,  lay— \-\'kx(d'- x)  into  —  (r+3— 4m). 


•<sC®^>- 


SECTION  Y 


DIVISION. 


Art.  114.  JIN  multiplication,  we  have  two  factors  given, 
and  are  required  to  find  their  product.  By  multiplying  the 
factors  4  and  6,  we  obtain  the  product  24.  But  it  is  fre- 
quently necessary  to  reverse  this  process.  The  number 
^4,  and  one  of  the  factors  may  be  given,  to  enable  us  to  find 
the  other.  The  operation  by  which  this  is  effected,  is  called 
Division,  We  obtain  the  number  4,  by  dividing  24  by  6. 
The  quantity  to  be  divided  is  called  the  dividend  ;  the  given 
factor,  the  divisor  ;  and  that  which  is  required,  the  quotient. 

115.  Division  is  finding  a  quotient,  which  multiplied 
iXTO  the  divisor  will  produce  the  dividend.* 

*  The  remainder  is  here  supposed  to  bo  included  in  the  qiiotielit,  as  is 
commonly  the  tase  in  algebra. 

7 


42  ALGEBRA. 

In  multiplication,  the  7nultiplier  is  always  a  iiumbtr*  (Art. 
91.)  And  i\\Q,  product  is  a  quantity  of  the  same  kind,  as  the 
multiplicand.  (Art.  92.)  The  product  of  3  rods  into  4,  i^ 
12  rods.  When  we  come  to  division,  the  product  and  either 
of  the  factors  may  he  given,  to  find  the  other  :  that  is, 

The  divisor  may  be  a  mimber,  and  then  the  quotient  will 
be  a  quantity  of  the  same  kind  as  the  dividend ;  or 

The  divisor  may  be  a  quantity  of  the  same  kind  as  the  div- 
idend ;  and  then  the  quotient  will  be  a  number. 

Thus  12  rods-^  4  =  3  rods.  But  12  rods -^3  rods  =4 

And  1 2  rods-r-  24  =^  rod.  And    1 2  rods-h-  24  rods=l: 

In  the  first  case,  the  divisor,  being  a  number,  shows  into 
ho7o  many  parts  the  dividend  is  to  be  separated  ;  and  the  quo- 
tient shows  what  these  parts  are. 

If  12  rods  be  divided  into  3  parts,  each  will  be  4  rods  long. 
And  if  12  rods  be  divided  into  24  parts,  each  will  be  half  a 
rod  long. 

In  tlie  other  case,  if  the  divisor  is  less  than  the  dividend, 
the  former  shows  into  what  parts  the  latter  is  to  be  divided ; 
and  the  quotient  shows  how  many  of  these  parts  are  contain- 
ed in  the  dividend.  In  other  words,  division  in  this  case 
consists  in  finding  how  often  one  quantity  is  contained  in  an- 
other. 

Aline  of  3  rods,  is  contained  in  one  of  12  rods,  four  times. 
'  But  if  the  divisor  is  greater  than  the  dividend,  and  yet  a 
quantity  of  the  same  kind,  the  quotient  shows  what  part  of 
the  divisor  is  equal  to  the  dividend. 

Thus  one  half  of  24  rods  is  equal  to  12  rods. 

116.  As  the  product  of  the  divisor  and  quotient  is  equal  to 
the  dividend,  the  quotient  may  be  found,  by  resolving  the 
dividend  into  two  such  factors,  that  one  of  them  shall  be  the 
divisor.     The  other  will  of  course,  be  the  quotient. 

Suppose  abd  is  to  be  divided  by  a.  The  factors  a  and  hd 
will  produce  the  dividend.  The  first  of  these,  being  a  divi- 
sor, may  be  set  aside.     The  other  is  the  quotient.     Hence, 

When  the  divisor  is  found  as  a  factor,  in  the  divi- 
dend, THE  division  IS  PERFORMED,  BY  CANCELLING  THIS  FAC- 
TOR. 


Divic 

lecx 

dh 

drx 

hmy 

dhxy 

abed 

abxy 

By 

c 

d 

dr 

hm 

dy 

b 

ax 

Quot.    X  X  hx  by 


DIVISION.  ^5 

In  each  of  these  examples,  the  letters  which  are  common 
to  the  divisor  and  dividend,  are  set  aside,  and  the  other  let- 
ters form  the  quotient.  It  will  be  seen  at  once,  that  the 
product  of  the  quotient  and  divisor  is  equal  to  the  dividend. 

117.  If  a  letter  is  repeated  in  the  dividend,  care  must  be 
taken  tliat  the  factor  rejected  be  only  equal  to  the  divisor* 
Div.    aab       bbx       aadddx       aammyy         aaaxxxh        yyy 
By      a  b  ad  amy  aaxx  yy 

Quot.  ab  ',      addx  ^  ^"         axh 


In  such  instances,  it  is  obvious  that  we  are  not  to  reject 
every  letter  in  the  dividend  which  is  the  same  with  one  in 
the  divisor. 

118.  If  the  dividend  consists  of  any  factors  whatever,  ex- 
punging one  of  them  is  dividing  by  it. 

Div.  a{b-\-d)     a{b-\-d)         [b-\-x){c-ird)       {b-^y)x{d-h)x 
By       a  b-{-d  b-^x  d—h 

Quot.  6  +  <?  a  c-f-t?  {b-\-y)xx 


In  all  these  instances  the  product  of  the  quotient  and  divi- 
sor is  equal  to  the  dividend  by  Art.  111. 

119.  In  performing  multiplication,  if  the  factors  contain 
numeral  figures^  these  are  multiplied  into  each  other.  (Art. 
96.)  Thus  Sa  into  lb  is  2\ab*  Now  if  this  process  is  to  be 
reversed^  it  is  evident  that  dividing  the  number  in  the  pro- 
duct, by  the  number  in  one  of  the  factors,  will  give  the  numr 
ber  in  the  other  factor.  The  quotient  of  21«6-^3ais  lb. 
Hence, 

In  division,  if  there  are  numeral  co-efficients  prefixed  to  the 
letters,  the  co-efficient  of  the  dividend  must  be  divided^  by  the 
co-efficient  of  the  divisor, 

Div.     6ff6  Udxy       25dhr       Uxy  34drx         ^Ohm 

By        2b  4dx  dh  6  34  m 

Quot.    3a  ^  /  25r  ,.  drx 

« 
120.  When  a  simple  factor  is  multiplied  into  a  compound 
one,  the  former  enters  into  every  term  of  the  latter.  (Art. 
98.)     Thus  a  into  b+d,  is  ab  +  ad.     Such  a  product  is  easi- 
ly resolved  again  into  its  original  factors. 


4^  ALGEBRA. 

Thus  ah  +  « J=«  xih-^-d) 

amh  4-  amx  +  amy = am  X  (A + ^  +^) 

4af?+ 8a^4- 1 2am+ 402/ =4a  X  (df-f  2^-f  3m+^) 

Now  if  the  whole  quantity  be  divided  by  one  of  these  fac- 
tors, according  to  Art.  118,  the  quotient  will  be  the  other 
factor. 

Thus    {ch-\-ad)-^a=^h-\-d.       And  (a6+a(Z)-r-(6  +  cif)=a. 
Hence, 

If  the  divisor  is  contained  in  every  term  of  a  compound  div- 
idend, it  must  be  cancelled  in  each, 

Div.     ab-\-ac       hdh-^-hdy        aah+ay  drx-^dhx  +  dxy 

By        a  hd  a  dx 

Quot.    h+c  i  -^  ff  ah+y  /   -*    ^    " 


And  if  there  are   co- efficients,  these  must  be  divided,  in 
each  term  also. 

Div.     6a6+12ac       10dry+16d  12hx  +  S       35dm+14dx 

By        3a                      2d  4                   7d 

Quot.    2b  -f  4c           J  *^  'f  -^  i'  SAoj  -f  2     -^'  '"  '  ■ 


121.  On  the  other  hand,  if  a  compound  expression  contain- 
ing any  factor  in  every  term,  be  divided  by  the  other  quantities 
connected  by  their  signs,  the  quotient  will  be  that  factor.  See 
the  first  part  of  the  preceding  article. 

Div.    ab-\-ac+ah    amh-+amx+amy  4ab  +  Zay     ahm-\-ahy 
Dy         h-\-C'\-h         h-^-x-^-y  h-\-2y  m-\-y 

Quot.  a  4a 


i  22.  In  division,  as  well  as  in  multiplication,  the  caution 
must  be  observed,  not  to  confound  terms  with  factors.  See 
Art.  99. 

Thus  (ah-\-ac)-^a=b-\'^c.     (Art.  120.) 

But     {abxac)-^a—aabc-^a=abc. 

And     iab-\-ac)-^(b+c)=a.     (Art.  121.) 

But     (abxac)-^{bxc)-=aabc'^bcz=aa. 


DIVISION.  45 

1 23.  In  division,  the  same  rule  is  to  be  observed  respec- 
ring  the  signs,  as  in  multiplication;  that  is,  if  the 
divisor  and  dividend  are  both  positive,  or  both  nega- 
tive, the  quotient  must  be  positive  i  if  one  is  positive 
and  the  other  negative,  the  quotient  must  be  negative. 
(Art.  105.) 

This  is  manifest  from  the  consideration  that  the  product  of 
the  divisor  and  quotient  must  be  the  same  as  the  dividend. 

—aX  +b  =  —ab  f    .i         J  —ab-^-^-b^^—a 

-{-aX  —b  —  '-ab  f  J   —ab-. b  =  -\-a 

—aX—b^-^ab  j  ^  -{-ab-z b=:--'a 

Div.      abx         8«— 10«y       3«^— 6«i/       Gamxdh 
By         —a         —2a  3a  —2a 


Quot.    —bx         —44-5?/        ^^  ^"'^  C         -3mxdh=i-  3hdm 

124.  If  THE  LETTERS  OF  THE  DIVISOR  ARE  NOT  TO  BE 
FOUND  IN  THE  DIVIDEND,  THE  DIVISION  IS  EXPRESSED  BY  WRI- 
TING THE  DIVISOR  UNDER  THE  DIVIDEND,  IN  THE  FORM  OP  A 
VULGAR  FRACTION. 

Xt/  -  d—x 

Thus  07^-4- a = — ;  ajid{d—x)~ — A= — r — 

This  is  a  method  of  denoting  division,  rather  than  an  actu- 
al performing  of  the  operation.  But  the  purposes  of  division 
iDay  frequently  be  answered,  by  these  fractional  expressions. 
As  they  are  of  the  same  nature  with  other  vulgar  fractions, 
they  may  be  added,  subtracted,  multiplied,  &c.  See  the  next 
Section. 

125.  When  the  dividend  is  a  compound  quantity,  the  di- 
visor may  either  be  placed  under  the  whole  dividend,  as  in  the 
preceding  instances,  or  it  may  be  repeated  under  each  term, 
taken  separately.  There  are  occasions  when  it  will  be  con- 
venient to  exchange  one  of  these  forms  of  expression  for  the 
other. 

m                     •    •                     .      .          b-\-c         b       c 
Thus  b-\-c  divided  by  x,  is  either ,  or— -f  ~7. 

X  XX 

a-^b 
And  « -ft  divided  by  2,  is  either— ^-  that  is,  half    the 


46  ALGEBRA. 


b 


«um  of  c  and  h\  Gf~^+^  ^^at  is,  the  sum  of  half  u  and 

half  b.     For  it  is  evident  that  half  the  sum  of  two  or  more 

quantities,  is  equal  to  the  sum  of  their  halves.     And  the  same 

principle  is  applicable,  to  a  third,  fourth,  fifth,  or  any  other 

portion  of  the  dividend. 

a—b         a       h 
So  also  a—h  divided  by  2,  is  either—^,  or  ~^—~^, 

For  half  the  difference  of  two  quantities,  is  equal  to  the 
difference  of  their  halves. 

a—2b+h       a      %      h  3a— c         Sa        c 

So    r = — —  —  4- — .   And = — 

>»  m      m  ^  m  —x        —x      —x. 

12G.  If  some  of  the  letters  in  the  divisor  are  in  each  term 
of  the  dividend,  the  fractional  expression  may  be  rendered 
more  simple,  by  rejecting  equal  factors  from  the  numerator 
and  denominator. 


Div.     ah            dhx 
By       ac            dy 

ahm—Say 
ah 

ah-\-bx 
by 

2am 
2xy 

ah      h 
Quot.  ac^iV 

hm—Sy 
b 

am 

These  reductions  are  made  upon  the  principle,  that  a 
given  divisor  is  contained  in  a  given  dividend,  just  as  many 
times,  as  double  the  divisor  in  double  the  dividend ;  triple 
the  divisor  in  triple  the  dividend,  <J/-c.  See  the  reduction  of 
fractions. 

127.  If  the  divisor  is  in  some  of  the  terms  of  the  dividend, 
but  not  in  all ;  those  which  contain  the  divisor  may  be  divi- 
ded as  in  Art.  116,  and  the  others  set  down  in  the  form  of  a 
fraction. 

ah-\-d       ah       d  d 

Thus  (fl6  4-  <?) -r-  a  is  either — - — ,  or      + — or  i + — • 

Div.    dxy-\-rx—hd         2ah-\-ad'\-x         bm-^-Sy       ^my-^dk 
By       a;  a  —6  2m 

hd  .  3y  ^ 

Qnot  dy+r-—  -^+1T 


DIVISION.  47 

128.  The  quotient  of  any  quantity  divided  by  itself  or  its 
equal,  is  obviously  a  unit. 

a  3ax  6  a+b—Sh 

Thus  -=1.  And3^=l.  And|q:^=l.  And^q:|:::3^=l. 

Div.    ax'\-x       Sbd—Sd      ^axi/—4a+Sad        Safe  4-3— 6m 
By       X  3d  4a  3 

Quot.  a+l         ^^  xy—l+^d 


Cor.  If  the  dividend  is  greater  than  the  divisor,  the  quo- 
tient must  be  greater  than  a  unit :  But  if  the  dividend  is  less 
than  the  divisor,  the  quotient  must  be  less  than  a  unit, 

PROMISCUOUS  EXAMPLES. 

1.  Divide  12£%+6«6a7— 186fem+246,  by  66. 

2.  Divide  16a  — 12-i-8?/+4  — 20aJa;-fm,  by  4. 

3.  Divide  (a—SA)  x  (3m+«/)  X  x,  by  (a— 2A)  X  (3m +3^). 

4.  Divide  ahd—^ad-^-Say  —  a^hyhd—Ad-^-Sy-^l,^^^ 

5.  Divide  ax—ry-\-ad—4my—Q-\-a,  by  —a. 

6.  Divide  a?rt_y + 3my — mx?/  4-  awi  —  c?,  by  —  dmy • 

7.  Divide  arc?— 6a+2r— M+6,by  2arJ. 

8.  Divide  6ax  — 8  +  2a:?/+4— 6Ay,  by  4aa?y. 

129.  From  the  nature  of  division  it  is  evident,  that  the 
value  of  the  quotient  depends  both  on  the  divisor  and  the 
dividend.  With  a  given  divisor,  the  greater  the  dividend^ 
the  greater  the  quotient.  And  with  a  given  dividend,  tlie 
greater  the  divisor,  the  less  the  quotient.  In  several  of  the 
succeeding  parts  of  algebra,  particularly  the  subjects  of 
fractions,  ratios,  and  proportion,  it  will  be  important  to  be 
able  to  determine  what  change  will  be  produced  in  the  quo- 
tient, by  increasing  or  diminishing  either  the  divisor  or  the 
dividend. 

If  the  given  dividend  be  24,  and  the  divisor  6  ;  the  quo- 
tient will  be  4.  But  this  same  dividend  may  be  supposed  to 
be  multiplied  or  divided  by  some  other  number,  before  it  is 
divided  by  6.  Or  the  divisor  may  be  multiplied  or  divided 
by  some  other  number,  before  it  is  used  in  dividing  24.  In 
each  of  these  cases,  the  quotient  will  be  altered. 

130.  In  the  first  place,  if  the  given  divisor  is  contained  in 
the  given  dividend  a  certain  number  of  times,  it  is  obvious; 
that  the  same  divisor  is  contained, 

In  doxfhh  that  dividend,  time  as  many  times ; 


48  ALGEBRA. 

In  triple  the  dividend,  thrice  as  many  times,  &c. 

That  is,  if  the  divisor  remains  the  same,  multiplying  the 
dividend  by  any  quantity,  is,  in  effect,  midtiplying  the  quotient 
by  that  quantity. 

Thus,  if  the  constant  divisor  is  6,  then  24—6=4  the  quotient- 
Multiplying  the  dividend  by  2,  2  x  24—  6=2x4 

Multiplying  by  any  number  n  w  X  24—  6 =n  X  4 

131.  Secondly,  if  the  given  divisor  is  contained  in  the  giv- 
en dividend  a  certain  number  of  times,  the  same  divisor  is 
contained, 

In  half  tliat  dividend,  half  as  many  times  ; 

In  one  third  of  the  dividend,  one  third  as  many  times,  &c. 

That  is,  if  the  divisor  remains  the  same,  dividing  the  divi- 
dend by  any  other  quantity,  is,  in  effect,  dividing  the  quotient 
by  that  quantity. 

Thus  24-^-6=4       , 

Dividing  the  dividend  by  2,  i24-^6=|4 

Dividing  by  9i,  j24H-6=,i4.   ' 

132.  Thirdly,  if  the  given  divisor  is  contained  in  the  giv^ 
en  dividend  a  certain  number  of  times,  then,  in  the  same 
dividend, 

Twice  that  divisor  is  contained  only  half  as  many  times ; 

Three  times  the  divisor  is  contained,  one  third  as  many  times. 

That  is,  if  the  dividend  remains  the  same,  multiplying  the 
divisor  by  any  quantity,  is,  in  effect,  dividing  the  quotient  by 
that  quantity. 

Thus  24-j-6=4 

Multiplying  the  divisor  by  2,  24-r  2  x  6  =f 

Multiplying  by  n  244- nxG^'^j,' 

133.  Lastly,  if  the  given  divisor  is  contained  in  the  given 
dividend  a  certain  number  of  times,  then,  in  the  same  divi- 
dend, 

Half  that  divisor  is  contained,  twice  as  many  times.. 

One  third  of  the  divisor  is  contained  thrice  as  many  times  ; 

That  is,  if  the  dividend  remains  tlie  same,  dividing  the  di- 
visor by  any  other  quantity,  is,  in  effect,  multiplying  the  quo- 
tient by  that  quantity. 

Thus  244-6=4 

Dividing  the  divisor  by  2,  24-^  i6 = 2  X  4 

Dividing  by  n,  244-  ^6  =n  X  4 

For  the  method  of  performing  division,  when  the  divisor 
and  dividend  are  both  compou7id  quantities^  see  one  of  the 
following  sections. 


SECTION  V 


FRACTIONS.^ 


Art.  134.  EXPRESSIONS  in  the  form  of  fractions  oc- 
cur more  frequently  in  algebra  than  in  arithmetic.  Most  in- 
stances in  division  belong  to  this  class.  Indeed  the  numera- 
tor of  every  fraction  may  be  considered  as  a  dividend^  of 
which  the  denominator  is  a  divisor. 

According  to  the  common  definition  in  aritlimetic,  the 
denominator  shows  into  what  parts  an  integral  unit  is  suppo- 
sed to  be  divided  ;  and  the  numerator  shows  how  many  of 
these  parts  belong  to  the  fraction.  But  it  makes  no  differ- 
ence, whether  the  whole  of  the  numerator  is  divided  by  the 
denominator;  or  only  one  of  the  integral  units  is  divided, 
and  then  the  quotient  taken  as  many  times,  as  the  number  of 
units  in  the  numerator.  Thus  |  is  the  same  as  i+i+i*  A 
fourth  part  of  three  dollars,  is  equal  to  three  fourths  of  one 
dollar. 

135.  The  value  of  a  fraction,  is  the  quotient  of  the  nume-^ 
rator  divided  by  the  denominator. 

6  ab 

Thus  the  value  of  ^  is  3.     The  value  of  -r  is  a. 

From  this  it  is  evident,  that  whatever  changes  are  made  in 
the  terms  of  a  fraction ;  if  the  quotient  is  not  altered,  the 
value  remains  the  same.  For  any  fraction,  therefore,  we 
may  substitute  any  other  fraction  which  will  give  the  same 
quotient. 

4      10     Aba     8drx     6-f2 

^"^  ¥=T=26^==45;j=3TI^''*  For  the  quotient 
in  each  of  these  instances  is  2. 

136.  As  the  value  of  a  fraction  is  the  quotient  of  the  nu- 
merator divided  by  the  denominator,  it  is  evident,  from  Art. 
128,  that  when  the  numerator  is  equal  to  the  denominator, 
the  value  of  the  fraction  is  a  unit ;  when  the  numerator  is 

*  Horsley's  Mathematics,  CamiilSi'  Aritllinetjc,  Einergon,  Euler,  Saun- 
derson,  and  Luulam. 
8 


50  ALGEBRA. 

less  than  the  denominator,  the  value  is  less  them  a  unit ;  and 
when  the  numerator  is  greater  than  the  denominator,  the 
vahie  is  greater  than  a  unit, 

The  calculations  in  fractions  depend  on  a  few  general 
principles,  which  will  here  be  stated  in  connection  with  each 
other. 

137.  If  the  denominator  of  a  fraction  remains  the  same"* 
multiplying  the  numerator  hy  any  quantity^  is  multiplying  the 
VALUE  by  that  quantity  ^  and  dividing  the  numerator^  is  divi- 
ding the  value.  For  the  numerator  and  denominator  are  a 
dividend  and  divisor,  of  which  the  value  of  the  fraction  is 
the  quotient.  And  by  Art.  130  and  1 31,  multiplying  the  div- 
idend is  in  effect  multiplying  the  quotient,  and  dividing  the 
dividend  is  dividing  the  quotient. 

ah    Sab     labd    ^ah 

Thus,  in  the  fractions — , , ,  ,  &c. 

'  a  ^     a  ^      a    ^     a  ^ 

The  quotients  or  values  are  6,      3^,    7bd,    ^h,  &;c. 

Here  it  will  be  seen  that,  while  the  denominator  is  not  al- 
tered, the  value  of  the  fraction  is  multiplied  or  divided  by 
the  same  quantity  as  the  numerator. 

Cor.  With  a  given  denominator,  the  greater  the  numera- 
tor, the  greater  will  be  the  value  of  the  fraction  ;  and,  on  the 
other  hand,  the  greater  the  value,  the  greater  the  numer- 
ator. 

138.  If  the  numerator  remains  the  same,  multiplying  the  de- 
7iominator  hy  any  quantity,  is  dividing  the  value  by  that  quan- 
tity /  a7id  dividing  the  denominator,  is  multiplying  the  value. 
For  multiplying  the  divisor  is  dividing  the  quotient ;  and 
dividing  the  divisor  is  multiplying  the  quotient.  (Art.  1 32, 
133.) 

24aZ»      24fl6      24«6      24a& 
In  the  fractions  -g^,    j^,    -^,     -j-,  (fee. 

The  values  are    4a,        2a,         8a,       24a,   <J-c. 

Cor.  With  a  given  numerator,  the  greater  the  denomina^ 
tor,  the  less  will  be  the  value  of  the  fraction  ;  and  the  less- 
the  value,  the  greater  the  denominator. 

139.  From  the  two  last  articles  it  follows,  that  dividing  the 
numerator  by  any  quantity,  will  have  the  same  effect  on  the 
value  of  the  fraction,  as  multiplying  the  denominator  by  that 
quantity  ;  and  multiplying  the  numerator  will  have  the  §ame 
effect,  as  dividing  the  denominator. 


FRACTIONS.  5i 

140.  It  is  also  evident,  from  the  preceding  articles,  that 

TF  THE  NUMERATOR  AND  DENOMINATOR  BE  BOTH  MULTIPLIED, 
OR  BOTH  DIVIDED,  BY  THE  SAME  QUANTITY,  THE  VALUE  OF 
THE  FRACTION  WILL  NOT  BE  ALTERED. 

bx     abx     3bx     }bx     ^abx 

these  instances  the  quotient  is  x\ 

141.  Any  integral  quantity  may,  without  altering  its  val- 
ue, be  thrown  into  the  form  of  a  fraction,  by  multiplying  the 
quantity  into  the  proposed  denominator,  and  taking  the  pro- 
duct for  a  numerator. 

a      ab     ad-\-ah     6adh 
Thus  a=Y=y=-^^=-g^,&c.    For  the  quotient 

of  each  of  these  is  a. 

dx  +  hx                            2drr+2dr 
So  d+h^ .     And  r-f  1  = r^ . 

1 42.  There  is  nothing  perhaps,  in  the  calculation  of  alge- 
braic fractions,  which  occasions  more  perplexity  to  a  learn- 
er, than  the  positive  and  negative  signs.  The  changes  in 
these  are  so  frequent,  that  it  is  necessary  to  become  familiar 
with  the  principles  on  which  they  are  made.  The  use  of 
the  sign  which  is  prefixed  to  the  dividing  line,  is  to  show 
whether  the  value  of  the  whole  fraction  is  to  be  added  to,  or 
subtracted  from,  the  other  quantities  with  which  it  is  con- 
nected. (Art.  43.)  This  sign,  therefore,  has  an  influence  on 
the  several  teiins  taken  collectively.  But  in  the  numerator 
and  denominator,  each  sign  affects  only  the  single  term  to 
which  it  is  apphed. 

ab 
The  value  of  -^  is  a,    (Art.  135.)    But  this  will  become 

negative,  if  the  sign  —  be  prefixed  to  the  fraction. 

a6  ab 

Thus  ^+-y=^  +  a.     But^-y =!/-«. 

So  that  changing  the  sign  which  is  before  the  whole  frac- 
tion, has  the  effect  of  changing  the  value  from  positive  to 
negative,  or  from  negative  to  positive. 

Next,  suppose  the  sign  or  signs  of  the  numerator  to  be 
changed. 

ab  -^ab 


By  Art.  123,  y=:-fr^.         But- 


b    - 


«, 


52  ALGEBRA. 

ah— he                                 ^ah+bc 
And — 7 — =4-«~c.  But 7 rr—or-fc. 

That  is,  by  changing  all  the  signs  of  the  numerator,  the 
value  of  the  fraction  is  changed  from  positive  to  negative,  or 
the  contrary. 

Again,  suppose  the  sign  of  the  denominator  to  be  changed. 

«6  ah 

As  before  -T-  =  -f-tt.  But3T  =  — «. 

143.  We  have,  then,  this  general  proposition;  If  the 

SIGN  PREFIXED  TO  A  FRACTION,  OR  ALL  THE  SIGNS  OF  THE  NU- 
MERATOR, OR  ALL  THE  SIGNS  OF  THE  DENOMINATOR  BE  CHANG- 
ED ;  THE  VALUE  OF  THE  FRACTION  WILL  BE  CHANGED,  FROM 
POSITIVE  TO  NEGATIVE,  OR  FROM  NEGATIVE  TO  POSITIVE. 

From  this  is  derived  another  important  principle.  As 
each  of  the  changes  mentioned  here  is  from  positive  to  neg- 
ative, or  the  contrary ;  if  any  izi^o  of  them  be  made  at  the 
same  time,  they  will  balance  each  other. 

Thus,  by  changing  the  sign  of  the  numerator, 

ah  —ah 

-^ =4. a  becomes ~7 — =— «. 

But,  by  changing  both  the  numerator  and  denominator,  it 
—ah 
becomes  '3^  =  + a?  where  the  positive  value  is  restored. 

By  changing  the  sign  before  the  fraction, 
ah  ah 

y+~l^'=^y+ct  becomes  ?/— y=^— «• 

But,  by  changing  the  sign  of  the  numerator  also,  it  be- 

comes  y— — 7~  where  the  quotient  —a  is  to  he  subtracted 

from  y,  or  which  is  the  same  thing,  (Art.  81,)  +a  is  to  be 
€tdded,  making  ^H- a  as  at  first.     Hence, 

144,  If  all  the  signs  both  of  the  numerator  and  de- 
nominator, OR  THE  SIGNS  OF  ONE  OF  THESE  WITH  THE  SIGN 
PREFIXED  TO  THE  WHOLE  FRACTION,  BE  CHANGED  AT  THE 
SAME  TIME,  THE  VALUE  OF  THE  FRACTION  WILL  NOT  BE  AL- 
TERED. 


FRACTIONS.  53 

6       -.6  -6  6 

6        -6  6  -6 

And  32=~2~=~~'2"~'~"— 2~~"^' 

Hence  the  quotient  in  division  may  be  set  down  in  differ- 

a      —c        a       c 
ent  ways.     Thus  (a— c)-^-^,  is  either  "T+'T":  ^^X~X* 

The  latter  method  is  the  most  common.  See  the  exam- 
ples in  Art.  127. 

REDUCTION  OF  FRACTIONS. 

145.  From  the  principles  which  have  been  stated,  are  de- 
rived the  rules  for  the  Reduction  of  fractions,  which  are  sub- 
stantially the  same  in  algebra,  as  in  arithmetic. 

A  FRACTION  MAY  BE  REDUCED  TO  LOWER  TERMS,  BY  DIVI- 
DING BOTH  THE  NUMERATOR  AND  DENOMINATOR,  BY  ANY  QUAN- 
TITY   WHICH    WILL    DIVIDE    THEM     WITHOUT     A    REMAINDER. 

According  to  Art.  140,  this  will  not  alter  the  value  of  the 
fraction. 

ab      a  Qdm     3m  7m        1 

Thus— r= — .     And-7rj-=~7-.     Andz — = — . 
CO      c  Bdi/      Ay  Imr       r 

In  the  last  example,  both  parts  of  the  fraction  are  divided 
by  the  numerator. 

a-{-hc  1  am-\-ay       a 

If  a  letter  is  in  every  term  both  of  the  numerator  and  de- 
nominator, it  may  be  cancelled^  for  this  is  dividing  by  that 
letter.    (Art.  120.) 

Sam-^-ay     Sm+y  dry+dy     r-fl 

^^^^ ~ad-{-ah   ="j+7i      ^"^  dhy-dy  =h^V 

If  the  numerator  and  denominator  be  divided  by  the 
greatest  common  measure,  it  is  evident  that  the  fraction  will 
be  reduced  to  the  lowest  terms.  For  the  method  of  finding 
the  greatest  common  measure,  see  Sec.  xvi. 

146.  Fractions  of  different  denominators  may  be  re- 
duced TO  A  COMMON  DENOMINATOR,  BY  MULTIPLYING  EACH 
NUMERATOR  INTO  ALL  THE  DENOMINATORS  EXCEPT  ITS  OWX, 


54  ALGEBRA. 

FOR    A    NEW    NUMERATOR  ;    AND    ALL    THE  DENOBIINATORS  TO- 
GETHER, FOR  A   COMMON  DENOMINATOR. 

Ex.  1.  Reduce-7-,  and -^,  and "--  to  a  common  denoniin- 

ator. 

dxdxyt^ady   ) 

cxb>:y=i cby    >    th e  three  numerators, 

mxbxdz=:mhdj 

hxdx y=bdy       the  common  denominator. 

.  ady  bey  bdm 

rhe  fractions  reduced  are  j^,  and  j^-,  and  j^. 

Here  it  will  be  seen,  that  the  reduction  consists  in  multi- 
plying the  numerator  and  denominator  of  each  fraction,  into 
all  the  other  denominators.  This  does  not  alter  the  value, 
(Art*  140.) 

dr  2k  Sc 

2.  Reduce  ^,  and  — ,  and  y. 

2  a  r+ 1 

3.  Reduce  —^-,  and ,  and  irrT. 

1  1 

4.  Reduce  "ttXi  and r. 

After  the  fractions  have  been  reduced  to  a  common  de- 
mominator,  they  may  be  brought  to  lower  terms,  by  the  rule 
in  the  last  article,  if  there  is  any  quantity,  which  will  divide 
the  denominator,  and  all  the  numerators,  without  a  remain- 
der* 

An  integer  and  a  fraction  are  easily  reduced  to  a  common 
denominator.     (Art.  141.) 

b  a  b         ttc  b 

Thus  a  and  —  are  equal  to  t-  and  — ,  or  —  and  -— . 

h      d  amy    bmy      hy        dm 

And  a,  b,  — .  —  are  equal  to ,  — - ,    "-,     ~~~ . 

^    ^  ni'    y  ^  my^my^my^     my 

147.  To  REDUCE  AN  IMPROPER  FRACTION  TO  A  MIXED 
QUANTITY,  DIVIDE  THE  NUMERATOR  BY  THE   DENOMINATOR,  aS 

in  Art.  127. 

ab-^-bm-^d  d 

Thus ^ =«-i-m-fy» 


FRACTIONS. 

am—a-^ady—hr 
Reduce ,  to  a  mixed  quantity. 


54 


For  the  reduction  of  a  mixed  quantity  to  an  improper 
fraction,  see  Art.  150,  And  for  the  reduction  of  a  compmmd 
fraction  to  a  simple  one,  see  Art.  160. 

ADDmON  OF  FRACTIONS. 

148.  In  adding  fractions,  we  may  either  write  tbem  one 
after  the  other,  with  their  signs,  as  in  the  addition  of  inte- 
gers, or  we  may  incorporate  them  into  a  single  fraction,  by 
the  following  rule  : 

Reduce  the  fractions  to  a  common  denominator,  make 
the  signs  before  them  all  positive,  and  then  add  their 
numerators. 

The  common  denominator  shows  into  what  parts  the  inte- 
gral unit  is  supposed  to  be  divided  ;  and  the  numerators  show 
the  number  of  these  parts  belonging  to  each  of  the  fractions. 
(Art.  134.)  Therefore  the  numerators  fftA;cn  ^oge/Acr  show 
the  whole  number  of  parts  in  all  the  fractions. 

^211  3111 

Thusy=yH-Y  And  y=y-|-y+Y, 

.,,.23111115 

Therefore  y-f  y=y-t-y4-  y-l-y+y=y. 

The  numerators  are  added,  according  to  the  rules  for  the 
addition  of  integers.  (Art.  69,  &c.)  It  is  obvious  that  the 
sum  is  to  be  placed  overiihe  common  denominator.  To  a- 
void  the  perplexity  which  might  be  occasioned  by  the  signs, 
it  will  be  expedient  to  make  those  prefixed  to  the  fractions 
uniformly  positive.  But  in  doing  this,  care  must  be  taken 
not  to  alter  the  value.  This  will  be  preserved,  if  all  the 
signs  m  the  numerator,  are  changed  at  the  same  time  with 
that  before  the  fraction.     (Art.  144.) 

2  4  2+4        6 

Ex.  1 .  Add  —  and  jT  of  a  pound.    Ans.  yg-  or  Tg* 

It  is  as  evident  that  ^g ,  and  tV  of  a  pound,  are  j\  of  a 
pound,  as  that  2  ounces,  and  4  ounces,  are  6  ounces. 

a  c 

2.  Add  ~f^  and  -g.  First  reduce  them  to  a  common  denom- 


56  ALGEBRA. 

ad         he  ctd-^bc 

inator.    They  will  then  be  ^  and  ^,  and  their  sum — ^j-, 

m  2r+J 

3.  Given  -g  and  —    ^^   ,  to  find  their  sum. 

m              2r-\-d     3hm             2dr^dd     Shm-^dr—dd. 
Ans.  -J  and  —^fr=^~3dh  ^^^  "  ~3^r"= 3dh 

a  h—m      a      —h-^-m      ay—hd-\-dm 

4.  Tand =-T+ =^— — j . 

d  y  d^       y  dy 

a  d        —am       dy        ^am-k-dy        am—dy 

5.  — and = 4- == or . 

y  —m      ^my      —my         —my  my 

a  h        aa—ab-^ab+hb     aa-\-bb 

^'  H^ ^"^ ^^b^aa+ab—ab-bh^aa-bb'     ^^^^'  ^^'^ 

—  a        —h  —4      —16 

7.  Add -^  to  ^^..      8.  Add -5- to  y^.     Ans.  —6. 

149.  For  many  purposes,  it  is  sufficient  to  add  fractions  in 
the  same  manner  as  integers  are  added,  by  writing  them  one 
after  another  with  their  signs.     (Art.  69.) 

a  3  d  a       3       d 

Thus  the  sum  of  -7-  and  ~and  —  ir— ,  is  -7-+ — —f^, 
0  y  2m  ^       0*   y      2m 

In  the  same  manner,  fractions  and  integers  may  be  added. 

The  sum  of  a  and  -—  and  3m  and  — — ,  is  a+3wi4- — -- — . 
y  r '  y     f 

150.  Or  the  integer  may  be  inco^rporated  with  the  fraction, 
by  converting  the  former  into  a  fraction,  and  then  adding  the 
numerators.     See  Art.  141. 

b         a      b      am     b      am-\-h 

The  sum  of  a  and  — ^  is  —-t- — =  — -f — = 

m'       I    ^  m      m      m         m 

rm  .    ^        h-\-d       3dm—3dy'\-h-\-d 

The  sum  of  3a  and ,  is 

m—y^  m—y 

Incorporating  an  integer  with  a  fraction,  is  the  same  as 
redoing  a  mixed  quantity  to  an  improper  fraction.  For  a 
mixed  quantity  is  an  integer  and  a  fraction.  In  arithmetic, 
these  are  generally  placed  together,  without  any  sign  be- 


FRACTIONS.  57 

tween  them.  But  in  algebra,  they  are  distinct  terms.  Thus 
2  i  is  2  and  i,  which  is  the  same  as  2+|. 

Ex.  1.  Reduce  «-|--7-  to  an  improper  fraction.     Ans. — r — ^.* 

r  hm—drn-^dh—dd  —  r 

2.  Reduce  m-\-d—  2~J*     ^^^-  h^^ * 

d  b  +  d  k 

3.  Reduce  1  +  T".     Ans. -r— .     4.  Reduce  1— ~« 

c  2d— 4: 

5.  Reduce  6+-^—-.     6.  Reduce  3+    ^^    » 

SUBTRACTION  OF  FRACTIONS. 

151 4  The  methods  of  performing  subtraction  in  algebra, 
depend  on  the  principle,  that  adding  a  negative  quantity  is 
equivalent  to  subtracting  a  positive  one  ;  and  r.  z;.  (Art.  81.) 
For  the  subtraction  of  fractions,  then,  we  have  the  following 
simple  rule.     Change  the  fraction  to  be  subtracted, 

FROM    positive     TO    NEGATIVE,   OR   THE  CONTRARY,  AND  THEN 

PROCEED  AS  IN  ADDITION.  (Art.  148.)  In  making  the  requir- 
ed change,  it  will  be  expedient  to  alter,  in  some  instances,  the 
signs  of  the  numerator,  and  in  others,  the  sign  before  the  di- 
viding line,  (Art.  143,)  so  as  to  leave  the  latter  always  affirm- 
ative. 

a  k 

Ex.  1.  From-7-,  subtra(it — , 
b  '  m 

h                                                             -/i 
First  change — ,  the  fraction  to  be  subtracted,  to • 

Secondly,  reduce  the  two  fractions  to  a  common  denomin- 

am  —bh 

ator,  making  -7 —  and  -7 — • 

Thirdly,  the  sum  of  the  numerators  am  —  6A,  placed  over 

am—bk 
the  common  denominator,  gives  the  answer,         — 7 . 

a-{-y  h  ad-^-dy—hr 

2.  From ,  subtract -y.       Ans. 7; . 

_  a  d—b  ay—dm  +  bm 

3.  From-  subtract— .      Ans.  — . 


^8  ALGEBRA. 

a+3d  3«— 2c/  17d—9a 

4.  From  — —  ,  subtract  — ^ •    -^^s.  — y^ — . 

b  —  d  b  by'-dy-\-bm 

5.  From  -^  subtract  -  -.      Ans- — . 

«+l  d-l  T.  3  4 

6.  From  -T~  subtract -——.     7.  From  —  subtract -r  • 

152.  Fractions  may  also  be  subtracted,  like  integers,  by 
setting  them  down,  after  their  signs  are  changed,  without  re- 
ducing them  to  a  common  denominator. 

h  h+d  h      h+d 

From  -  subtract  -— .     Ans.  ■^  +— • 

In  the  same  manner,  an  integer  may  be  subtracted  from  a 
fraction,  or  a  fraction  from  an  integer. 

b  b 

From  a  subtract  — .       Ans.  «  — ~r- 

153.  Or  the  integer  maybe  incorporated  with  the  frac- 
tion, as  in  Art.  150.   • 

h                                  k            h—my 
Ex.  1.  From — subtract  »x.     Ans. --— m= . 

y  y  y 

b^  h  acd+bd+hc 

2.  From  4«-t-—^  subtract  3«—  -r.     Ans. ^ . 

b—c  c—b  cZ+26— 2c 

3.  From  1+     1  ■   subtract      j    .     Ans. -3 —    . 

f?— 6  d+b 

3.  From  a -I- 3A—-^—  subtract  3a— A+--r-. 

MULTIPLICATION  OF  FRACTIONS. 

154.  By  the  definition  of  multiplication,  multiplying  by  a 
fraction  is  taking  a  part  of  the  multiplicand,  as  many  times, 
as  there  are  like  parts  of  an  unit  in  the  multiplier.  (Art.  90.) 
Now  the  denominator  of  a  fraction  shows  into  what  parts  the 
integral  unit  is  supposed  to  be  divided  ;  and  the  numerator 
shows  how  many  of  those  parts  belong  to  the  given  fraction. 
In  multiplying  by  a  fraction,  therefore,  the  multiplicand  is  to 
be  divided  into  such  parts,  as  are  denoted  by  the  denomina- 
tor ;  and  then  one  of  these  parts  is  to  be  repeated,  as  many 
times,  as  is  required  by  the  numerator. 


FRACTIONS,  ^9 

3 

Suppose  «  is  to  be  multiplied  byr. 

A  fourth  part  of  a  is  -—. 

a      a       a      Sa 
This  taken  3  times  is  4'+T  + V^T*  ^^^**  ^^^'^ 

a  3 

Again,  suppose  -r  is  to  be  multiplied  by~. 

a  a 


One  fourth  of  y  is  77.   (Art.  138.) 

«       a      a       3 


a       a      a       3a 
This  taken  3  times  is  7I+77;+7I=-TI 


the  product  required. 

In  a  similar  manner,  any  fractional  multiplicand  may  be 
divided  into  parts,  by  multiplying  the  denominator  ;  and  one 
of  the  parts  may  be  repeated,  by  multiplying  the  numerator. 
We  have  then  the  foUomng  rule : 

155.  To  MULTIPLY  FRACTIONS,  MULTIPLY  THE  NUMERA- 
TORS TOGETHER,  FOR  A  NEW  NUMERATOR,  AND  THE  DENO- 
MINATORS   TOGETHER,    FOR    A    NEW    DENOMINATOR. 

3b  d  3hd 

Ex.  1.  Multiply—  into  ^.    Product  ^. 

a-\-d  Ah  Aah-\'Adh 

2.  Multiply  into ^.     Product  -— -~, 

^ '^      y  m—2  my--2y 

{a-\-m)xh  4  {a  +  m)Xih 

3.  Multiply  ^ i^to-^^3;;-^  Product -^^^^^. 

a  +  h  4— m  1  3 

156.  1^'  ihe  meth<^  multiplying  is  the  same,  when  there 
are  '^*  inore  than  two  ^.j^j^g  ^^  jjg  multiphed  together. 

1.  Multiply  tog^_  ~  andy.     Product  ^. 

a        c   .  i  ac  m     acm 

^^^  T^l  '^'  ^^  V  article  ^,  and  this  into  y  is  j^^- 

,  ,  .  ,    2^  ^             1                       '2abh-2abd 
1.  Multiply—:,    ,.\,and — r.    Product   ^ . 


60  ALGEBRA. 

3.  Mult.  -^,  y  and  ^:^,    4.  Mult.  ^,  jj^^,  and  -y . 

157.  The  multiplication  may  sometimes  be  shortened,  by 
rejecting  equal  factors,  from  the  numerators  and  denomi- 
nators. 

.        a  .         A  d  dh 

1.  Multiply  —  into — and — .     Product  — . 

^  •'    ?'  a  y  ry 

Here  a  being  in  one  of  the  numerators,  and  in  one  of  the 
denominators,  may  be  omitted.     If  it  be  retained,  the  pro- 
adh 

duct  will  be .     But  this  reduced  to   lower  terms,  by 

ary  '     -^ 

dh 

Art.  145,  will  become  —  as  before. 
ry 

ad  m         ah  ah 

2.  Multiply  —  into  —  and  ^.     Product  -^. 

It  is  necessary  that  the  factors  rejected  from  the  numera- 
tors be  exactly  equal  to  those  which  are  rejected  from  the 
denominators.  In  the  last  example,  a  being  in  two  of  the 
numerators,  and  in  only  one  of  the  denominators,  must  be 
retained  in  one  of  the  numerators. 

a-\-d  my  am-\-dm 

.  3.  Multiply  into  — r.     Product  — ^ — 

Here,  though  the  same  letter  a  is  in  one  of  the  numera- 
tors, and  in  one  of  the  denominators,  yet  as  it  is  not  in  every 
term  of  the  numerator,  it  must  not  be  cancelled. 

am-\-d  h  3r 

4.  Multiply  -J-  into  -  and  -. 

If  any  difficulty  is  found,  in  makin^^®^  ^^  •*' -ontractions,  it 
will  be  better  to  perform  the  multipP"'  withoM  t-  -^omitting 
any  of  the  factors  5  and  to  reduce  th^^^^^  ^^  lower  i^ru  onns 
afterwards.  "^^  ^** 

158.  When  a  fraction  and  an  .^^  are  multiplied  to- 
gether, the  numerator  of  the  frac  *^  i^ultiplied  into  the 
integer.  The  denominator  is  n*^^^^  5  except  in  cases 
where  division  of  the  denominate  ^"^^^^^^^ed  for  multipli- 
cation of  the  numerator,  accorc^^  ^^^'  ^  ^^* 


FRACTIONS.  (-| 

m     am  a  a      m      am 

Thus«x — = — .     For  «=—;  and  — X  — = — . 

y     y  1         ^     y     y 

X      A4-1      hrx  +  rx  1       a 

^^  ''^'d^1~='~^r~-  Andax-^=y.     Hence, 

159.  A  FRACTION  IS  MULTIPLIED  INTO  A  QUANTITY  EQUAL 
TO  ITS  DENOMINATOR,  BY  CANCELLING   THE  DENOMINATOR. 

«  a  ah  I 

Thus  y  X  6  =flf.  For  y  x  6  =  y .  But  the  letter  b,  be- 
ing in  both  the  numerator  and  denominator,  may  be  set 
aside.     (Art.  145.) 

3m  h-\-3d 

So  ^i:^x(a-?/)=3m.     And  ^T^X  (3+m)=A+3</. 

On  the  same  principle,  a  fraction  is  multiplied  into  any 
factor  in  its  denoninator,  by  cancelling  that  factor. 

a  ay      a  h  h 

160.  From  the  definition  of  multiplication  by  a  fraction, 
it  follows  that  what  is  commonly  called  a  compound  fraction,^ 

3         a 
is  the  product  of  two  or  more  fractions.     Thus  —  of  -7-  is 

3       «  3        a  I       « 

-j-X  -7".     For  -7-  of -7-,  is  y  of -7-  taken  three  times,  that  is, 

a       a       a  a  3 

TF+TT  +  TT.     But  this  is  the  same  as  -7-  multiplied  by  — . 

(Art.  154.) 

Hence,  reducing  a  compound  fraction  to  a  simple  one,  is  the 
same,  as  multiplying  fractions  into  each  other, 

Ex.   1.  Reduce  -ir  of  ,-r7r.     Ans. 


7  "'   b  +  2'     """°"  76+14- 

2        4          b  +  h  Sh  +  Sh 

2.  Reduce  IT  of  T"  of  t; .     Ans. 


3        5         2a— m*  30a  — 15m* 

1         1  1  1 

3.  Reduce  y  of  y  of  y:^^-.     Ans.  igg-Sl^* 

*  By  a  compound  fraction  is  meant  a  fraction  of  a  fraction,  and  not  a 
traction  whose  numerator  or  denominator  is  a  compound  quantity. 


(y2  ALGEBRA. 

161.  The  expressions  fa,  |&,  4y?  ^^»  are  equivalent  lo 
— ,  — ,  -y ,     For  fa  is  |  of  a,  which  is  equal  to  -r-  Xfl= 

y •     (Art.  1 58.)     So  ih  =|  X  ^=  y . 

,. DIVISION  OF  FRACTIONS. 

162.  To  DIVIDE  ONE  FRACTION  BY  ANOTHER,  INVERT  THE 
DIVISOR,  AND  THEN  PROCEED  AS  IN  MULTIPLICATION.  (Art. 
155.) 

a         c  a      d      ad 

Ex.  I.   Divide  yby-j.    Ans.yX7=j^. 

To  understand  the  reason  of  the  rule,  let  it  be  premised, 
that  the  product  of  any  fraction  into  the  same  fraction  invert- 
ed is  always  a  unit. 

a       h      ah  r.    .     d       h+y 

'"^'^T>^T=^  =  ^-     ^''^h+^''d=''     (Art.  128.) 

But  a  quantity  is  not  altered  by  multiplying  it  by  a  unit. 
Therefore  if  a  dividend  be  multiphed,  first  into  the  divisor 
inverted,  and  then  into  the  divisor  itself,  the  last  product 
will  be  equal  to  the  dividend.  Now,  by  the  definition,  art. 
115,  "  division  is  finding  a  quotient,  which  multiphed  into 
the  divisor  will  produce  the  dividend."  And  as  the  dividend 
multiphed  into  the  divisor  inverted  is  such  a  quantity,  the 
quotient  is  truly  found  by  the  rule. 

This  explanation  will  probably  be  best  understood,  by  at- 
tending to  the  examples.  In  several  which  follow,  the  jjroof 
of  the  division  will  be  given,  by  multiplying  the  quotient  in- 
to the  divisor.  This  will  present,  at  one  view,  the  dividend 
multiplied  into  the  inverted  divisor,  and  into  the  divisor  it- 
self, 

m        Sk  m      y      my 

3.  Divide  25  by  - .     Ans.  ^^th=^dh' 

my      Sk     m 
P'°°^-  6dh^J=Td  *^  dividend. 

x-{-d      od  x-\-d      y       xyxdy 

3.  Divide by —     Ans. X~:rj'= — 73 — • 

r       ^  y  r        od         5ar 

xy+dy     5d    x  +  d 
Proof.-^/x-=— . 


FRACTIONS.  ^3 

Adh        4hr  4dh       a       ad 

4.  Divide  --  by  - -.    Ans.  —  X  j^-=-. 

ad     Ahr     4dh 

Proof.  — X — = —  the  dividend. 

rx       a         X 

36d       ISh  Sed     lOy     4% 

5.  Divide  —  by  Y^-.       Ans.  --  Xj^^=-j^. 

^  ab-\'l        ab  —  l  ^  A— my  3 

6.  Divide  — ^ —  by .     7.  Divide  — :; —  by  rTT. 

163.  When  a  fraction  is  divided  by  an  integer,  the  denom- 
inator of  the  fraction  is  multiphed  into  the  integer. 

Thus  the  quotient  of-T~  divided  by  w,  is  t—  . 

m  a       m      a       I        a 

For  m =-^ ;  and  by  the  last  article,  ~L-^~r  ="r  X  —  =7- . 

1  1  1  1  3  3       1 

So T-rrh=^ 7Xt~  =  ~7 7T«     And  ""T-r 6  =  ^  =:-r", 

a—b  a  —  b      h      ah  —  bh  4  24      8 

In  fractions,  multiplication  is  made  to  perform  the  office 
of  division ;  because  division  in  the  usual  form  often  leaves 
a  troublesome  remainder  :  but  there  is  no  remainder  in  mul- 
tiplication. In  many  cases,  there  are  methods  of  shortening 
the  operation-  But  these  will  be  suggested  by  practice, 
without  the  aid  of  particular  rules. 

164.  By  the  definition,  art.  49,  "  the  reciprocal  of  a  quan- 
tity, is  the  quotient  arising  from  dividing  a  unit  by  that 
-quantity." 

a  a  b     h 

Therefore,  the  reciprocal  of -p,  is  1  -r-r-= 1  x  ""  =  -That  is, 

The  reciprocal  of  a  fraction  is  the  fraction  inverted^ 

b  m+v  1 

Thus  tlie  reciprocal  of  — ]-— is  — t—  ;  the  reciprocal  of  ^ 

% 
is  -T-  or  3y ;  the  reciprocal  of  |  is  4.     Hence  the  reciprocal 

of  a  fraction  whose  numerator  is  1 ,  is  the  denominator  of 
the  fraction. 

Thus  the  reciprocal  of—  is  « ;  of  -—^r,  is  a+b,  kc. 


54  ALGEBRA. 

165.     A  fraction  sometimes  occurs  in  the  numerator  or  de- 


a 


nominator  of  another  fraction,  as  -y-.    It  is  often  convenient, 

in  the  course  of  a  calculation,  to  transfer  such  a  fraction, 
from  the  numerator  to  the  denominator  of  the  principal 
fraction,  or  the  contrary.  That  this  may  be  done,  without 
altering  the  value,  if  the  fraction  transferred  be  inverted^  is 
evident,  from  the  following  principles  : 

First,  Dividing  by  a  fraction,  is  the  same  as  multiplying  by 
the  fraction  inverted,     (Art.  162.) 

Secondly,  ^Dividing  the  numerator  of  a  fraction  has  the 
same  effect  on  the  value,  as  multiplying  the  denominator  ;  and 
multiplying  the  numerator  has  the  same  effect,  as  dividing 
the  denominator.     (Art.  139.) 

fa  a    .  . 

Thus  in  the  expression  —  the  numerator  of —   is    multi- 

'^  X  X 

plied  into  f .     But  the  value  will  be  the  same,  if,  instead  of 
multiplying  the  numerator,  we  divide  the  denominator  by  ^ 
that  is,  multiply  the  denominator  by 


5  5 
5 
3 


f«       a  h      ^h 

Therefore  —  =  7-.  So  —  = — . 

a?      fx  Im      m 

166.  Multiplying  the  numerator  is  in  effect  multiplying  the 
value  of  the  fraction.  (Art.  137.)  On  this  principle,  a  frac- 
tion may  be  cleared  of  a  fractional  co-efficient  which  occurs 
in  its  numerator. 

Ja      3       a      3«  \a      I       a       a 

Thus -7-=— XT  =71-     And— =— X — =7-. 

0      5      0     5b  y      ^      y     ^y 

\h-{-\x        1       h-\-x     h-\-x  fa?       3a; 

^"d    ""^r~"=l"><~^=~3;^-    ^"^^  =  20^- 

3«      3       a      fa 

On  the  other  hand,  — =—  x  — =^-^. 
'  7a;      7       x       x 

a       \       a      la  4a  f« 

167.  But  multiplying  the  denominator,  by  another  fraction, 
is  in  effect  dividing  the  value  ;  (Art.  138.)  that  is,  it  is  multi- 
plying the  value  by  the  fraction  inverted.  The  principal 
fraction  may  therefore  be  cleared  of  a  fractional  co-efficient, 
which  occurs  in  its  denominator. 


SIMPLE  EQUATIONS. 

^       a       5 

5a 

And  : 

«      7« 

|5~2S" 

3h 
4m 

2U 

"  4m' 

On  the  other  hand, 

7a 
3a;' 

a 

Sy'\'3dx 
2m 

y-{-dx 

And 

3aj      a? 

y  ^¥j 

6. J 


167.&.  The  numerator  or  the  denominator  of  a  fraction, 
may  he  itself  a  fraction.  The  expression  may  be  reduced 
to  a  more  simple  form,  on  the  principles  which  have  been 
applied  in  the  preceding  cases. 

a 

h        a       c      ad 


Thus 


c         b   '    d      be 
1 


X 


y  X  r        nr 

And  — r-  =  7—.  And = — . 

ti        ny  mm 


•«C®^3 


SECTION  VII. 


SIMPLE  EQUATIONS. 

Art.  168.  X  he  subjects  of  the  preceding  sections  are 
introductory  to  what  may  be  considered  the  peculiar  pro- 
vince of  algebra,  the  investigation  of  the  values  of  unknown 
quantities,  by  means  of  equations. 

An  equation  is  a  proposition,  expressing  in  algebraic 

CHARACTERS,  THE  EQUALITY  BETWEEN  ONE  QUANTITY  OR  SET 
10 


66  ALGEBKA. 

OF  QUANTITIES  AND  ANOTHER,  OR  BETWEEN  DIFFERENT  EX* 
PRESSIONS  FOR  THE  SAME  QUANTITY.*       ThuS  CC-f  a=5  +  C,  IS 

an  equation,  in  which  the  sum  of  x  and  a,  is  equal  to  the  sum 
of  b  and  c.  The  quantities  on  the  two  sides  of  the  sign  of 
oquahty,  are  sometimes  called  the  members  of  the  equation ; 
the  several  terms  on  the  left  constituting  the  Jirst  member, 
and  those  on  the  right,  the  second  member. 

169.  The  object  aimed  at,  in  what  is  called  the  resolution 
or  reduction  of  an  equation,  is  to  find  the  value  of  the  un- 
known quantity/.  In  the  first  statement  of  the  conditions  of 
a  problem,  the  known  and  unknown  quantities  are  frequent- 
ly thrown  promiscuously  together.  To  find  the  value  of 
that  which  is  required,  it  is  necessary  to  bring  it  to  stand  by 
itself,  while  all  the  others  are  on  the  opposite  side  of  the 
equation.  But,  in  doing  this,  care  must  be  taken  not  to  de- 
stroy the  equation,  by  rendering  the  two  members  unequal. 
Many  changes  may  be  made  in  the  arrangement  of  the 
terms,  without  affecting  the  equality  of  the  sides. 

170.  The  reduction  of  an  equation  consists,  then,  in 
bringing  the  unknown  quantity  by  itself,  on  one  side, 
and  all  the  known  quantities  on  the  other  side,  with- 
out destroying  the  equation. 

To  effect  this,  it  is  evident  that  one  of  the  members  must 
be  as  much  increased  or  diminished  as  the  other.  If  a  quan- 
tity be  added  to  one,  and  not  to  the  other,  the  equality  will 
be  destroyed.  But  the  members  will  remain  equal ; 
If  the  same  or  equal  quantities  be  added  to  each.  Ax.  1. 
If  the  same  or  equal  quantities  be  subtracted  from  each.  Ax.  2. 
If  each  be  multiplied  by  the  same  or  equal  quantities.  Ax.  3. 
If  each  be  divided  by  the  same  or  equal  quantities.  Ax.  4. 

171.  It  may  be  farther  observed  that,  in  general,  if  the 
unknown  quantity  is  connected  with  others  by  addition,  mul- 
tiplication, division,  &lc,  the  reduction  is  made  by  a  contrary 
process.  If  a  known  quantity  is  added  to  the  unknown,  the 
equation  is  reduced  by  subtraction.  If  one  is  multiplied  by 
the  other,  the  reduction  is  effected  by  division,  &c.  The 
reason  of  this  will  be  seen,  by  attending  to  the  several  cases 
in  the  following  articles.  The  knoian  quantities  may  be  ex- 
pressed either  by  letters  or  figures.  The  unknoivn  quantity 
is  represented  by  one  of  the  last  letters  of  the  alphabet,  gen- 
erally, X,  y,  or  z.     (Art.    27.)     The  principal  reductions  to 

*  See  Note  D. 


SIMPLE  EQUATIONS-  6  7 

be  considered  in  this  section,  are  those  which  are  effected 
by  transposition,  multiplication,  and  division.  These  ought 
to  be  made  perfectly  famiUar,  as  one  or  more  of  them  will 
be  necessary,  in  the  resolution  of  almost  every  equation. 

TRANSPOSITION. 

172.  In  the  equation 

the  number  7  being  connected  with  the  unknown  quanti- 
ty X  by  the  sign  — ,  the  one  is  subtracted  from  the  other.  To 
reduce  the  equation  by  a  contrary  process,  let  7  be  added  to 
both  sides.     It  then  becomes 

a;-7-f7  =  9  +  7. 
The  equality  of  the  members  is  preserved,  because  one  is  as 
much  increased  as  the  other.  (Axiom  1.)  But  on  one  side, 
we  have  —7  and  +7.  As  these  are  equal,  and  have  contra- 
ry signs,  they  balance  each  other,  and  may  be  cancelled.  (Art. 
77.)     The  equation  will  then  be 

a:  =  9-f7. 
Here  the  value  of  x  is  found.     It  is  shown  to  be  equal  to 
9-f7,  that  is  to  16.     The  equation  is  therefore  reduced. 
The  unknown  quantity  is  on  one  side  by  itself,  and  all  the 
known  quantities  on  the  other  side. 

In  the  same  manner,  if  x—b—a 

Adding  b  to  both  sides  x—h-^-b^a-^-b 

And  cancelling  {—b-\-b)  x-=a-\-b. 

Here  it  will  be  seen  that  the  last  equation  is  the  same  as 
the  first,  except  that  b  is  on  the  opposite  side,  with  a  contra- 
ry sign. 

N^xt  suppose  y-\-c=d. 

Here  c  is  added  to  the  unknown  quantity  ?/.  To  reduce  the 
equation  by  a  contrary  process,  let  c  be  subtracted  from  both 
sides,  that  is,  let  —  c  be  appUed  to  both  sides*     We  then  have 

y-^c—c=:d—c. 
The  equality  of  the  members  is  not  affected,  because  one  is 
as  much  diminished  as  the  other.     When  (  +  e— c)  is  cancel- 
led, the  equation  is  reduced,  and  is 

y^d—c. 
This  is  the  same   as    y-\-c^d,  except   that  c  has  been 
transposed,  and  has  received  a  contrary  sign.     We  hence 
obtain  the  following  general  rule : 


68  ALGEBRA. 

173.  When  known  quantities  are  connected  with  the 
unknown  quantity  by  the  sign  +  or  —,  the  equation  is 

reduced  by  TRANSPOSING  THE    KNOWN    QUANTITIES    TO    THE 
OTHER  SIDE,  AND  PREFIXING  THE  CONTRARY  SIGN. 

This  is  called  reducing  an  equation  by  addition  or  subtrac- 
tion, because  it  is,  in  effect,  adding  or  subtracting  certain 
quantities,  to  or  from,  each  of  the  members. 
Ex.  1.  Reduce  the  equation  a;  +  35— m=rA— c? 

Transposing  +3b,  we  have  cc—m^h-^d—Sh 

And  transposing —w,  a:=^— <Z--36+m 

1 74.  When  several  terms  on  the  same  side  of  an  equation 
are  alike,  they  may  be  united  in  one,  by  the  rules  for  reduc- 
tion in  addition.     (Art.  72  and  74.) 

Ex.  2.  Reduce  the  equation  a:+56— 4/t~76 

Transposing  5^— 4^  a:=7&— -56+4^ 

Uniting  76  —  56  in  one  term  a:=26+4A. 

175.  The  unknown  quantity  must  also  be  transposed, 
whenever  it  is  on  both  sides  of  the  equation.  It  is  not  ma- 
terial on  which  side  it  is  finally  placed.  For  if  ac  =  3  ;  it  is 
evident  that  3= a:.  It  may  be  well  however,  to  bring  it  on 
that  side,  where  it  will  have  the  affirmative  sign,  when  the 
equation  is  reduced. 

Ex.3.  Reduce  the  equation  ^x  +  ^h^h+d-^Sx 

By  transposition  2A  —  A — c?= 3a; — 2x 

And  h'-d:=x, 

1 76.  When  the  same  term,  with  the  same  sign,  is  on  opposite 
sides  of  the  equation,  instead  of  transposing,  we  may  expunge 
it  from  each.  For  this  is  only  subtracting  the*  same  quantity 
from  equal  quantities.     (Ax.  2.) 

Ex.  4.  Reduce  the  equation  cc-|-37i4-cZ=6  +  3/»4-7c? 

Expunging  3 A  x-^d^b+ld 

And  x=zb-{-6d. 

177.  As  all  the  terms  of  an  equation  may  be  transposed, 
or  supposed  to  be  transposed ;  and  it  is  immaterial  which 
sifiember  is  written  first ;  it  is  evident  that  the  signs  of  all  the 
terms  may  be  changed,  without  affecting  the  equality. 

Thus,  if  we  have  x—h—d^a 

Then  by  transposition  —  <?-}-ac=— x-f  6 

Or,  inverting  the  members  —a; +6  =  — €?-+-«• 

178.  If  all  the  terms  on  one  side  of  an  equation  be  trans- 
posed, each  member  will  be  equal  to  0. 


SIMPLE  EQUATIONS.  G  9 

Thus,  if  a;-4-i5>=(/,  then  x+h-'d^O. 

It  is  frequently  convenient  to  reduce  an  equation  to  this 
form,  in  which  the  positive  and  negative  terms  balance  each 
other.  In  the  example  just  given,  a;  4-^  is  balanced  by  —d. 
For  in  the  first  of  the  two  equations,  a; +6  is  equal  to  df. 

Ex.5.  Reduce  «-f  2a; —  8  =6  —  4-1-0? + a.    \    - 

6.  Reduce  iz-^-ab— hm= a -{-^y—ab  +  hm, 

7.  Reduce  h+S0-{-7x==S-eh-{-6x'-d-\-b.   X 

8.  Reduce  bh+2l—4x^d=:^2—Sx-\-d^7bh. 

REDUCTION  OF  EQUATIONS  BY  MULTIPLICATION. 

1 79.  The  unknown  quantity,  instead  of  being  connected 
with  a  known  quantity  by  the  sign  -j-  or  — ,  may  be  divided 

X 

by  it,  as  in  the  equation  —  =5. 

Here  the  reduction  can  not  be  made,  as  in  the  preceding 
instances,  by  transposition.  But  if  both  members  be  multi- 
plied by  «,  (Art.  1 70,)  the  equation  will  become 

xr=ab. 
For  a  fraction  is  multiplied  into  its  denominator^  by  remov^ 
ing  the  denominator.     This  has  been  proved  from  the  prop- 
erties of  fractions.    (Art.  159.)     It  is  also  evident  from  the 
sixth  axiom. 

ax     3a?      {a-\-b)xx     dx-k-lix    ^       ^  , 

Thus  a;= — =-^= — — rr — =■     i  ,  .   -,  &c.  For  in  each 
a       3  a-f6  ^+5   ' 

of  these  instances,  a;  is  both  multiplied  and  divided  by  the 

same  quantity ;  and  this  makes  no  alteration  in  the  value. 

Hence, 

180.  When  the  unknown  quantity  is  divided  by  a 
known  quantity,  the  equation  is  reduced  by  multiply- 
ing each  side  by  this  known  quantity. 

The  same  transpositions  are  to  be  made  in  this  case,  as  in 
the  preceding  examples.  It  must  be  observed  also,  that 
every  term  of  the  equation  is  to  be  multiplied.  For  the  sev- 
eral terms  in  each  member  constitute  a  compound  multipH- 
cand,  which  is  to  be  multiphed  according  to  art.  98. 


Ex.  1 .  Reduce  the  equation  —  -|-  a = i  -f-  c? 

Multiplying  both  sides  by  c 


The  product  is  x+ac^hc-^cd 

And  a;=6c+cc?-- flc. 


70  ALGEBRA. 

_  a;  —  4 

2.  Reduce  the  equation  —7— +5=20 

Multiplying  by  6  a:  -  4  +  30  =  1 20 

And  a;  =  120+4  — 30=94. 

X 

3.  Reduce  the  equation  ~*ZrA"^"^~^ 

Multiplying  by  «+&  (Art.  100.)  x+ad+hd=:=ah-\-bh 
And  x=ah'{'bh-'ad—bd» 

181.  When  the  unknown  quantity  is  in  the  denominator  of 
a  fraction,  the  reduction  is  made  in  a  similar  manner,  by  mul- 
tiplying the  equation  by  this  denominator. 

6 
Ex*  4.^  Reduce  the  equation  TnZir  +  7  —  8 

Multiplying  by  1 0 — a;  6  +  70 — Ta; = 80 — 8.t 

And  a;  =4. 

1 82.  Though  it  is  not  generally  necessary,  yet  it  is  often 
convenient,  to  remove  the  c'enominator  from  a  fraction  con- 
sisting of  known  quantities  only.  This  may  be  done,  in  the 
same  manner,  as  the  denominator  is  removed  from  a  fraction 
which  contains  the  unknown  quantity. 

_  ,     ^  ,  X       d       k 

Take  for  example  —  =  T~+ — 

^  a       b  ^  c 

ad    ah 
Multiplying  by  a  a:=-T"+ — 

Multiplying  by  &  6a;=«6?+ 

c 

Multiplying  by  c  bcx=acd-[-abh. 

Or  we  may  multiply  by  the  product  of  all  the  denomina 
iors  at  once. 

X       d       h 

In  the  same  equation  — =17  +  — 

^  a       0       c 

obex     abed      abch 
Multiplying  by  abc  —^ = — ^  +  —^ 

Then  by  cancelling  from  each  term,  the  letter  which  is 
common  to  its  numerator  and  denominator,  (Art,  1 45,)  we 
have  bcx=:acd-\-abh,  as  before.     Hence, 

183.  An  equation  xMay  be  cleared  of  fractions  by  mul- 
tiplying EACH  SIDE  INTO  ALL  THE  DENOMINATORS. 


SIMPLE  EQUATIONS.  7 1 

oc       J)       c       Jl 

Thus  the  equation  — =^+ — — — 

^  a       a      g      m 

Is  the  same  as  dgmx=:ahgm-\-adem—adgIu 

X       2       4      Q 

And  the  equation  9~~"T'^~5^"^^ 

Is  the  same  as  30a:  =40+ 48+ 180. 

In  clearing  an  equation  of  fractions,  it  will  he  necessary 
to  observe,  that  the  sign  —  prefixed  to  any  fraction,  denotes 
that  the  whole  value  is  to  be  subtracted,  (Art.  142,)  which  is 
done  by  changing  the  signs  of  all  the  terms  in  the  numera- 
tor. 

The  equation  =c— 7 

Is  the  same  as  ar^dr=:crx—3bx+2hmx-{'6nx* 

REDUCTION  OF  EQUATIONS  BY  DIVISION. 

184.  When  THE  unknown  quantity  is  multiplied  into 

ANY    known   quantity,  THE  EQUATION  IS  REDUCED  BY  DIVI- 
DING BOTH  SIDES  BY  THIS  KNOWN  QUANTITY.    (Ax,  4.) 

Ex.  1.  Reduce  the  equation  fla;+6  — 3A=£? 

By  transposition  ax=d+3k--h 

d-\-3h-^b 


Dividing  by  a 


xssr 


a 

a      d 

Reduce  the  equation  2a:  = —  —  -7"+ 4^ 

Clearing  of  fractions  2chx^=^ak'-cd-^4hch 

ah^cd+Abch 


Dividing  by  2c/t  a:=s- 


2ch 


185.  If  the  unknown  quantity  has  co-efficients  in  several 
terms,  tlie  equation  must  be  divided  by  all  these  co-efficients, 
connected  by  their  signs,  according  to  art.  121. 

Ex.  3.  Reduce  the  equation  3x—hx=a—d 

That  is,  (Art.  120.)  (3-6)  Xx-a-d 

a^d 
Dividing  by  3 ~Z>  a;=<,__7 

Ex.  4.  Reduce  the  equation  «x  +  a;=A— 4 

A-4 
Dividing  by  a+1  ^'==«+l 


72  x\X.QEeRA. 

Ex.  5.  Reduce^lie  equation  ^  "^  ""IT"  ^""^J"" 

Clearing  of  fractious  Ahx —iix=xak-{-dh'^b 

,  ah-^-dh—^b 

Dmding  by  4A— 4  0:=;- — Ah^n 

186.  If  any  quantity,  either  known  or  unknown,  is  found 
as  a  factor  in  every  term,  the  equation  may  be  divided  by  it» 
On  the  other  hand,  if  any  quantity  is  a  divisor  in  every  term, 
the  equation  may  be  multiplied  by  it,  In  this  way,  the  fac- 
tor o?  divisor  will  be  removed,  so  as  to  render  the  expression 
more  simple. 

Ex.6.  Reduce  the  equation  ax-\'3ab=:6ad-{-a 

Dividing  by  a  a:+36=6tZ-l-l 

And  a?=6J4-l— 35 

a?4-l      b      h-d 
7.  Reduce  the  equation  —-—=—— 

Multiplying  by  a?  (Art.  159.)  a?-|- 1  —b=k-d 

And  x=^h—d-{-b  —  U 

8 .  Reduce  the  equation  x  x  (c+ J) — « — 6 = rf  X  (a+6) 

Dividing  by  a+6  (Art.  1 1 8.)  a;  —  1  =£? 
And  x=id+l* 


187.  Sometimes  the  conditions  of  a  "problem  are  at  first 
stated,  not  in  an  equation,  but  by  means  of  sl  proportion.  To 
show  how  this  may  be  reduced  to  an  equation,  it  will  be  ne- 
cessary to  anticipate  the  subject  of  a  future  section,  so  far 
as  to  admit  the  principle  that  "  when  four  quantities  are  in 
geometrical  proportion,  the  product  of  the  two  extremes  is 
equal  to  the  product  of  the  two  means  :"  a  principle  which 
is  at  the  foundation  of  the  Rule  of  Three  in  arithmetic. 
See  Webber's  Arithmetic. 

Thus,  ii  a:b::c:d,  Then  ad=:hc 

AMif  3:4::6:8;  And3x8=4x6.     Hence, 

188.  A  PROPORTION  IS  CONVERTED  INTO  AN  EQUATION,  BT 
MAKING  T^E  PRODUCT  OF  THE  EXTREMES,  ONE  SIDE  OF  THE 
EQUATION  \  AND  THE  PRODUCT  OF  THE  MEANS>  THE  OTHER 
SIDE. 


SIMPLE  EQUATIONS.  73 

Ex.  U  Reduce  to  an  equation  ax:h::ch:d. 

The  product  of  the  extremes  is  adx 

The  product  of  the  means  is  bch 

The  equation  is,  therefore  adx=:hcJu 

2.  Reduce  to  an  equation  a+h  :c  I'lh—mi^/, 

The  equation  is  ay+by=ich—cm, 

189.  On  THE  OTHER  HAND,  AN  EQUATION  MAY  BE  CONVER- 
TED INTO  A  PROPORTION,  BY  RESOLVING  ONE  SIDE  OF  THE 
EQUATION  INTO  TWO  FACTORS,  FOR  THE  MIDDLE  TERMS  OF  THE 
PROPORTION  ;  AND  THE  OTHER  SIDE  INTO  TWO  FACTORS,  FOR 
THE  EXTREMES. 

As  a  quantity  may  often  be  resolved  into  different  pairs  of 
iactors  ;  (Art.  42,)  a  variety  of  proportions  may  frequently 
be  derived  from  the  same  equation. 

Ex.  1.  Reduce  to  a  proportion  abc-=^deh 

The  side  abc  may  be  resolved  into  a  x  &c,  or  abxc,  or  acxb. 
And  deh  may  be  resolved  into         dxeh,  or  dexh  or  dhx e. 
Therefore  aid::  eh:  be  And  ac  :dh: :  e:b 

And  ab:de::h:c  And  acidiieh:  b  ^rc. 

For  in  each  of  these  instances,  the  product  of  the  ex- 
tremes is  abc,  and  the  product  of  the  means  deh, 

2.  Reduce  to  a  proportion  ax-\-bx=cd—ch 

The  first  member  may  be  resolved  into  a:  x  («+6) 

And  the  second  into  cx(d—h) 

Therefore  :v:c: :  d—h  :  a-^-b        And  d — h  :  a  : :  a-f  b  : c.  Sic, 

190.  l(^  for  any  term  or  terms  in  an  equation,  any  other 

expression  of  the  same  value  be  substituted,  it  is  manifest 

that  the  equality  of  the  sides  will  not  be  affected. 

64 
Thus,  instead  of  16,  we  may  write  2x8  or~,  or  25—9,  <Src, 

For  these  are  only  different  forms  of  expression  for  the 
same  quantity. 

191.  It  will  generally  be  well  to  have  the  several  steps,  in 
the  reduction  of  equations,  succeed  each  other  in  the  follow- 
order. 

First,  Clear  the  equation  of  fractions.  (Art.  183.) 
Secondly,  Transpose  and  unite  the  terms.  (Arts.  173,  4,  5.) 
Thirdly,  Divide  by  the  co-eflOicieHts  of  the  unknown  quan- 
tity. (Arts.  184,  5.) 

n 


74  ALGEBRA. 

Examples, 

3x  bx 

1.  Reduce  the  equation  •  X^"^~"F  +  ^ 

Clearing  of  fractions  24a; +192= 20a;  4-224 

Transp.  and  uniting  terms  4a:  =  32 

Dividing  by  4  a?  =  8. 

a;  a;       a; 

2.  Reduce  the  equation  — 4-^=-7-— — 4-cZ 

Clearing  of  fractions  bcx-\-abx'-acx^=abcd—ahck 

abed  —  abck 

3.  Reduce  40— 6a;~16  =  120— 14a:,  Ans.  a;  =  12. 

a;— 3      X  a;-19  ^  9^3 

4.  Reduce —^+-^=20— — ^  Ans.a?=~. 

a?       a;  a  1— a 

5.  Reduce  y+ ^=20-— •       6.  Reduce -^-4=5. 

3  6a? 

7.  Reduce  rT7— 2=8.  8.  Reduce —73-:  =  1 . 

XX  X        X         X         1 

9.  Reduce  a; +"^ 4-"^= n.     10.  Reduce -2+"^— f =75- 

a?— 5  284— a? 

11.  Reduce  '—r--^Qx=. — . 

4  0 

2a?+6  1107-37 

12.  Reduce 3a7+ — r— =5+ — s, . 


14.  Reduce  214 


6a?— 4  18  — 4a? 

3a?-ll     5^-5     .97-7a; 


13.  Reduce — ^ — —2= — ^ -\-x. 


16     ~"     8     ^       2 


a?— 4  5a?4-l4       1 

15.  Reduce  3a:  — —j—— 4= — -^To- 

7a?+5      16  +  4a:  3a;+9 

16.  Reduce — ^ — — — 7 4-6= — ^ — • 

17-30!      4a?+2  7.r-fl4 

17.  Reduce ^ —  — — ^ — =5— 6a?4-" — ^ — • 

3Ar  — 3  20— a?     6;f— 8     4a:— 4 

18.  Reduce  a;— — ^-  +  4= — ^ — — — ij — +""5 — • 


19.  Reduce 


SIMPLE  EQUATIONS. 


9       '     Gx-^'S  3 

5x-\-4    18—0? 
20.  Reduce  — ^ —  •  — :; — : :  7  : 4. 


SOLUTION  OF  PROBLEMS. 

192,  In  the  solution  of  problems,  by  means  of  equations, 
two'^things  are  necessary :  First,  to  translate  the  statement 
of  the  question  from  common  to  algebraic  language,  in  such 
a  manner  as  to  form  an  equation  :  Secondly,  to  reduce  this 
equation  to  a  state  in  which  the  unknown  quantity  will  stand 
by  itself,  and  its  value  be  given  in  known  terms,  on  the  oppo- 
site side.  The  manner  in  which  the  latter  is  effected,  has 
already  been  considered.  The  former  will  probably  occasion 
more  perplexity  to  a  beginner  ;  because  the  conditions  of 
questions  are  so  various  in  their  nature,  that  the  proper  meth- 
od of  stating  them  cannot  be  easily  learned,  like  the  reduc- 
tion of  equations,  by  a  system  of  definite  rules.  Practice 
however  will  soon  remove  a  great  part  of  the  difficulty. 

193.  It  is  one  of  the  principal  peculiarities  of  an  algebra- 
ic solution,  that  the  quantity  sought  is  itself  introduced  into 
the  operation.  This  enables  us  to  make  a  statement  of  the 
conditions,  in  the  same  form,  as  though  the  problem  were  al- 
ready solved.  Nothing  then  remains  to  be  done,  but  to  re- 
duce the  equation,  and  to  find  the  aggregate  value  of  the 
known  quantities.  (Art.  53.)  As  these  are  equal  to  the  un- 
known quantity  on  the  other  side  of  the  equation,  the  value 
of  that  also  is  determined,  and  therefore  the  problem  is 
solved. 

Problem  1.  A  man  being  asked  how  much  he  gave  for  his 
watch,  replied ;  If  you  "multiply  the  price  by  4,  and  to  the 
product  add  70,  and  from  this  sum  subtract  50,  the  remain- 
der will  be  equal  to  220  dollars. 

To  solve  this,  we  must  first  translate  the  conditions  of  the 
problem,  into  such  algebraic  expressions,  as  will  form  an  e- 
quation. 

Let  the  price  of  the  watch  be  represented  by  x 
This  price  is  to  be  mult'd  by  4,  which  makes  4a? 
To  the  product,  70  is  to  be  added,  making  4a?-|-70 

From  this,  50  is  to  be  subtracted,  making  4a; -f- 70—50 


70  ALGEBRA. 

Here  we  have  a  number  of  the  conditions,  expressed  in 
algebraic  terms  ;  but  have  as  yet  no  equation.  We  must  ob- 
serve then,  that  by  the  last  condition  of  the  problem,  the 
preceding  terms  are  said  to  be  equal  to  220. 

We  have,  therefore,  this  equation  4a; +  70—50=220. 

Which  reduced  gives  x=50. 

Here  the  value  of  x  is  found  to  be  50  dollars,  which  is  the 
price  of  the  watch. 

1 94.  To  prove  whether  we  have  obtained  the  true  value 
of  the  letter  which  represents  the  unknown  quantity,  we  have 
only  to  substitute  this  value,  for  the  letter  itself,  in  the  equa- 
tion which  contains  the  first  statement  of  the  conditions  of 
the  problem ;  and  to  see  whether  the  sides  are  equal,  after 
the  substitution  is  made.  For  if  the  answer  thus  satisfies 
the  conditions  proposed,  it  is  the  quantity  sought.  Thus,  iji 
the  preceding  example, 

The  original  equation  is  4^+70—50=220 

Substituting  50  for  x,  it  becomes       4  X  50+70—50=220 
That  is,  220=220. 

Prob.  2.  What  number  is  that,  to  which,  if  its  half  be  ad- 
ded, and  from  the  sum  20  be  subtracted,  the  remainder  will 
be  a  fourth  part  of  the  number  itself? 

In  stating  questions  of  this  kind,  where  fractions  are  con- 
cerned, it  should  be  recollected,  that  ^x  is  the  same  as 
X  2t 

y  ;  that  ia:=y,  &c.     (Art.  161.) 

In  this  problem,  let  x  be  put  for  the  number  required. 

XX 

Then  by  the  conditions  proposed,  x-^-—  —20=— 

And  reducing  the  equation  a:  =16. 

^       .  16  16 

Proof  i6+Y-20=-j. 

Prob.  3.  A  father  divides  his  estate  among  his  three  sons, 
in  such  a  manner,  that. 

The  first  has  glOOO  less  than  half  of  the  whole  ; 
The  second  has  800  less  than  one  third  of  the  whole  ; 
The  third  has  600  less  than  a  fourth  of  the  whole  ; 
What  is  the  value  of  the  estate  ? 
If  the  whole  estate  be  represented  by  x,  then  the  several 

X  XX 

shares  will  be-y  —  lOOO,  and  -^  —800,  and  —  —600. 


SIMPLE  EQUATIONS.  g/  Tj. 

And  as  these  constitute  the  whole  estate,  they%re^icfith-o3Er 


■er  equal  to  a?. 

We  have  then  this  equation  ^--  1000+y-800+— -600=a:. 

Which  reduced  gives  ,t— 28800. 

28800  28800  28800 

Proof   —Y  - 1000+ -y"-800+-^— 600=28800. 

195.  To  avoid  an  unnecessary  introduction  of  unknown 
quantities  into  an  equation^  it  may  he  well  to  observe,  in  this 
place,  that  when  the  sum  or  difference  of  two  quantities  is 
given,  both  of  them  may  be  expressed  by  means  of  the 
same  letter.  For  if  one  of  the  two  quantities  be  subtracted 
from  their  sum,  it  is  evident  the  remainder  will  be  equal  to 
the  other.  And  if  the  difference  of  two  quantities  be  sub- 
tracted from  the  greater,  the  remainder  will  be  the  less. 

Thus,  if  the  sum  of  the  two  numbers  be  20 

And  if  one  of  them  be  represented  by  x 

The  other  will  be  equal  to  20 — x, 

Prob.  4.  Divide  48  into  two  such  parts,  that  if  the  less 
be  divided  by  4,  and  the  greater  by  6,  the  sum  of  the  quo- 
tients will  be  9. 

Here,  if  x  be  put  for  the  smaller  part,  the  greater  will  be 

48—07. 

X      48  —  0? 
By  the  conditions  of  the  problem  —-f-  — ^  =9. 

Therefore  a;  =  12,  the  less. 

And  48 — a: = 36,  the  greater. 

196.  Letters  may  be  employed  to  express  the  known  quan- 
tities in  an  equation,  as  well  as  the  unknown.  A  particular 
value  is  assigned  to  the  numbers,  when  they  are  introduced 
into  the  calculation :  and  at  the  close,  the  numbers  are  re- 
stored.    (Art.  52.) 

Prob.  5.  If  to  a  certain  number,  720  be  added,  and  the 
sum  be  divided  by  125  ;  the  quotient  will  be  equal  to  7392 
divided  by  462.     What  is  that  number  ? 

Let  x=  the  number  required. 

rt=720  J=7392 

/^  =  125  h-462 


^ 


78»  ALGEBRA. 

TIacn  by  the  conditions  of  the  problem  —  —  _. 

_  hd—ah 

There  to  re  .r = — -. — 

h 

(125X739'2)-(720X462) 
Restoringthe  numbers,  x— tjt^ =  1 280. 

197.  "When  the  resolution  of  an  equation  brings  out  a  ntg- 
fUive  answer,  it  shows  that  the  value  of  the  unknown  quanti- 
ty is  contrary  to  the  quantities  which,  in  the  statement  of  the 
question,  are  considered  positive.  See  Negative  Quantities. 
(Art.  54,  kc.) 

Prob.  6.  A  merchant  gains  or  loses,  in  a  bargain,  a  certain 
sum*  In  a  second  bargain,  he  gains  350  dollars,  and,  in  a 
third,  loses  60.  In  the  end,  he  finds  he  has  gained  200  dol- 
lars, by  the  three  together.  How  much  did  he  gain  or  lose 
by  the  first  ? 

In  this  example,  as  the  profit  and  loss  are  opposite  in  their 
nature,  they  must  be  distinguished  by  contrary  signs.     (Art. 
57.)     If  the  profit  is  marked  +,  the  loss  must  be  —. 
Let  a;=  the  sum  required. 

Then  according  to  the  statement  a; -[-350—60=200 

And  a;  =  —90 

The  negative  sign  prefixed  to  the  answer,  shows  that  there 
was  a  loss  in  the  first  bargain  ;  and  therefore  that  the  proper 
sign  of  X  is  negative  also.  But  this  being  determined  by  the 
answer,  the  omission  of  it  in  the  course  of  the  calculation 
<?an  lead  to  no  mistake. 

Prob.  7.  A  ship  sails  4  degrees  north,  then  13  S.  then  17 
N.  then  19  S.  and  has  finally  11  degrees  of  south  latitude. 
What  was  her  latitude  at  starting  ? 

Let  X  =  the  latitude  sought. 
Then  marking  the  northings  -f ,  and  the  southings  —  ; 
By  the  statement  a:  +  4-13-f  17-19=- U 

A]E>d  a;=0. 

Tlic  answer  here  shows  that  the  place  from  which  the  ship 
started  was  on  the  equator,  where  the  latitude  is  nothing. 

Prob.  8.  If  a  certain  number  is  divided  by  1 2,  the  quotient, 
dividend,  and  divisor  added  together,  will  amount  to  64- 
What  is  the  number  ? 


SIMPLE  EQUATIONS.  79 


Let  a?=  the  number  sought. 

Then 

~+X+\2=:G4 

And 

624 

Prob.  9.  An  estate  is  divided  among  four  children,  in  such 
a  manner,  that 

The  first  has  200  dollars  more  than  I  of  the  whole, 
The  second  has  340  dollars  more  than  |  of  the  whole, 
The  third  has  300  dollars  more  than  ^  of  the  whole. 
The  fourth  has  400  dollars  more  than  |  of  the  whole. 
What  is  the  value  of  the  estate  ?  Ans.  4800  dollars. 

Prob.  10.  What  is  that  number  which  is  as  much  less 
than  500,  as  a  fifth  part  of  it  is  greater  than  40  ?     Ans.  450. 

Prob.  11.  There  are  two  numbers  whose  difference  is  40, 
and  which  are  to  each  other  as  6  to  5.  What  are  the  num- 
bers ?  Ans.  240  and  200. 

Prob.  12.  Three  persons,  Jl,  B,  and  C  draw  prizes  in  a 
lottery.  ^  draws  200  dollars  ;  B  draws  as  much  as  ^,  to- 
gether with  a  third  of  what  C  draws  ;  and  C  draws  as  much 
as  A  and  B  both.     What  is  the  amount  of  the  three  prizes  ? 

Ans.   1200  dollars. 

Prob.  13.  What  number  is  that,  which  is  to  12  increased 
by  three  times  the  number,  as  2  to  9  ?  Ans.  8. 

Prob.  14.  A  ship  and  a  boat  are  descending  a  river  at  the 
same  time.  The  ship  passes  a  certain  fort,  when  the  boat  is 
13  miles  below.  The  ship  descends  five  miles,  while  the 
boat  descends  three.  At  what  distance  below  the  fort,  will 
they  be  together  ?  Ans.  32 1  miles. 

Prob.  15.  What  number  is  that,  a  sixth  part  of  which  ex- 
ceeds an  eighth  part  of  it  by  20  ?  Ans.  480. 

Prob.  16.  Divide  a  prize  of  2000  dollars  into  two  such 
parts,  that  one  of  them  shall  be  to  the  other,  as  9:7. 

Ans.  The  parts  are  1 125,  and  875. 

Prob.  1 7.  What  sum  of  money  is  that,  whose  third  part, 
fourth  part,  and  fifth  part,  added  together,  amount  to  94 
dollars?  ,  Ans.  120  dollars. 


go  ALGEBRA. 

Prob.  18.  Two  travellers,  A  and  B,  360  miles  apart,  trav- 
el towards  each  other  till  they  meet,  ^'s  progress  is  10 
miles  an  hour,  and  JB's  8.  How  far  does  each  travel  before 
they  meet?  Ans.  A  goes  200  miles,  and  B  160. 

Prob.  19.  A  man  spent  one  third  of  his  life  in  England, 
one  fourth  of  it  in  Scotland,  and  the  remainder  of  it,  which 
was  20  years,  in  the  United  States.  To  what  age  did  he 
live  ?  Ans.  To  the  age  of  48. 

Prob.  20.  What  number  is  that,  \  of  which  is  greater 
than  I  of  it  by  96  / 

Prob.  21.  A  post  is  ^  in  the  earth,  f  in  the  water,  and  13 
feet  above  the  water.     What  is  the  length  of  the  post  1 

Ans.  35  feet. 

prob.  22.  What  number  is  that,  to  which  10  being  added^ 
J  of  the  sum  will  be  ^  ? 

Prob.  23.  Of  the  trees  in  an  orchard,  f  are  apple  trees, 
Jjj  pear  trees,  and  the  remainder  peach  trees,  which  are  20 
more  than  \  of  the  whole.  What  is  the  whole  number  in 
the  orchard  ?  Ans.  800. 

Prob.  24.  A  gentleman  bought  several  gallons  of  wine 
for  94  dollars  ;  and  after  using  7  gallons  himself,  sold  \  of 
the  remainder  for  20  dollars.  How  many  gallons  had  he 
at  first  ?  Ans.  47. 

Prob.  25.  A  and  B  have  the  same  income.  A  contracts 
an  annual  debt  amounting  to  -}  of  it  ;  B  lives  upon  |-  of  it ; 
at  the  end  of  10  years,  B  lends  to  A  enough  to  pay  off  his 
debts,  and  has  160  dollars  to  spare.  What  is  the  income  of 
each  ?  Ans.  280  dollars. 

Prob.  26.  A  gentleman  lived  single  }  of  his  whole  life  ; 
and  after  having  been  married  5  years  more  than  4  of  his 
life,  he  had  a  son  who  died  4  years  before  him,  and  who 
reached  only  half  the  age  of  his  father.  To  what  age  did 
the  father  live  ?  Ans.  84. 

Prob.  27,  What  number  is  that,  to  which,  if  |,  |,  and  f  of 
it  be  added,  the  sum  will  be  73  ? 

Prob.  28.  A  person,  after  spending  100  dollars  more  than 
\  of  his  income^  had  remaining  35  dollars  more  than  v  of 
it.    Required  his  income. 


SIMPLE  EQUATIONS.  Ct 

Prob.  29.  In  the  composition  of  a  quantity  of  gimpowderj 
The  nitre  was  10  lb.  more  than  |  of  the  whole, 
The  sulpkiir  4J  lb.  less  than  ^  of  the  whole, 
The  charcoal  2  lb.  less  than  |  of  the  nitre. 

What  was  the  amount  of  gunpowder  ?     Ans.  69  lb. 

Prob.  30.  A  cask  which  held  146  gallons,  was  filled  with 
a  mixture  of  brandy,  wine,  and  water.  There  were  15  gal- 
lons of  wine  more  than  of  brandy,  and  as  much  water  as  the 
brandy  and  wine  together.  What  quantity  was  therb  of 
each  ?  \, 

Prob.  31.  Four  persons  purchased  a  farm  in  company  for 
4755  dollars ;  of  which  B  paid  three  times  as  much  as  A ; 
C  paid  as  much  as  A  and  B  ;  and  D  paid  as  much  as  C  and 
B,     What  did  each  pay  ?  Ans.  317,  951,  1268,  2219. 

Prob.  32.  It  is  required  to  divide  the  number  99  into  five 
such  parts,  that  the  first  may  exceed  the  second  by  3, 
be  less  than  the  third  by  1 0,  greater  than  the  fourth  by  9, 
and  less  than  the  fifth  by  16. 

Let  x=i  the  first  part. 
Then  a;  — 3=  the  second,  as— 9=  the  fourth, 

x4-10=  the  third,  a;+16=  the  fifth. 

Therefore  a;4-x--3-ha:+10+a;-9+a;4- 16=99. 

And  a;  =  17. 
Prob.  33.    A  father  divided  a  small  sum  among  four  sons. 
The  third  had  9  shillings  more  than  the  fourth ; 
The  second  had  1 2  shillings  more  than  the  third ; 
The  first  had  1 8  shillings  more  than  the  second ; 
And  the  whole  sum  was  6  shillings  more  than  7  times 

the  sum  which  the  youngest  received. 
What  was  the  sum  divided  ?  Ans.  \b3. 

Prob.  34.  A  farmer  had  two  flocks  of  sheep,  each  contain- 
ing the  same  number.  Having  sold  from  one  of  these  39, 
and  from  the  other  93,  he  finds  twice  as  many  remaining  in 
the  one,  as  in  the  other.  How  many  did  each  flock  origin- 
ally contain  ? 

Prob.  35.  An  express,  travelling  at  the  rate  of  60  miles  a* 
day,  had  been  dispatched  5  days,  when  a  second  was  sent  after 
him,  travelling  75  miles  a  day.  "  In  what  time  will  the  one 
overtake  the  other  ?  Ans.   20  days. 

Prob.  36      The  age  of  A  is  double  that  of  B,  the  age  of 

B  triple  that  of  C,  and  the  sum  of  ^W  their  a^cs  140.    What 

is  the  ag.Q  of  each  ? 

12 


82  ALGEBRA, 

Prob.  37.  Two  pieces  of  cloth,  of  the  same  price  by  the 
yard,  but  of  different  lengths,  were  bought,  the  one  for  five 
pounds,  the  other  for  6|.  If  10  be  added  to  the  length  of 
each,  the  sums  will  be  as  5  to  6.  Required  the  length  of 
each  piece.       <     *i    -  -^ 

Prob.  38.  A  and  B  began  trade  with  equal  sums  of  mon- 
ey. The  first  year,  A  gained  forty  pounds,  and  B  lost  40. 
The  second  year,  ^  lost  ^  of  what  he  had  at  the  end  of  the 
first,  and  B  gained  40  pounds  less  than  twice  the  sum  which 
A  had  lost.  B  had  then  twice  as  much  money  as  A.  What 
sum  did  each  begin  with  ?  Ans.  320  pounds. 

Prob.  39.  What  number  is  that,  which  being  severally  ad- 
ded to  36  and  52,  will  make  the  former  sum  to  the  latter,  as 
3  to  4  ? 

Prob.  40.  A  gentleman  bought  a  chaise,  horse,  and  har- 
ness, for  360  dollars.  The  horse  cost  twice  as  much  as  the 
harness ;  and  the  chaise  cost  twice  as  much  as  the  harness 
and  horse  together.     WTiat  was  the  price  of  each  ? 

Prob.  41.  Out  of  a  cask  of  wine,  from  which  had  leaked 
^  part,  21  gallons  were  afterwards  drawn;  when  the  cask 
was  found  to  be  half  full.     How  much  did  it  hold  ? 

Prob.  42.  A  man  has  6  sons,  each  of  whom  is  4  years 
older  than  his  next  younger  brother  ;  and  the  eldest  is  three 
times  as  old  aa  the  youngest.     What  is  the  age  of  each  ? 

Prob.  43.  Divide  the  number  49  into  two  such  parts, 
that  the  greater  increased  by  6,  shall  be  to  the  less  diminish- 
ed by  11,  as  9  to  2. 

Prob.  44.  What  two  numbers  are  as  2  to  3 ;  to  each  of 
which,  if  4  be  added,  the  sums  will  be  as  5  to  7  ? 

Prob.  45.  A  person  bought  two  casks  of  porter,  one  of 
which  held  just  3  times  as  much  as  the  other;  from  each  of 
these  he  drew  4  gallons,  and  then  found  that  there  were  4 
times  as  many  gallons  remaining  in  the  larger,  as  in  the  oth- 
er.    How  many  gallons  were  there  in  each  ? 

Prob.  46.  Divide  the  number  68  into  two  such  parts,  that 
the  difference  between  the  greater  and  84,  shall  be  equal  to 
3  times  the  difference  between  the  less  and  40. 

Prob.  47.  Four  places  are  situated  in  the  order  of  the 
letters  A,  B.  C,  D,  The  distance  from  A  to  D,  is  34  miles. 
The  distance  from  A  to  B  is  to  the  distance  from  C  to  D  as 
2  to  3.     And  i  of  the  distance  from  A  to  B,  added  to  half 


POWERS.  83 

the  distance  from  C  to  D,  is  three  times  the  distance  from 
B  to  C.     What  are  the  respective  distances  ? 
Ans.  From^to5=12;  fromBtoC=4;  fromCtoZ)=18. 

Prob.  48.  Divide  the  number  36  into  3  such  parts,  that  ^ 
of  the  first,  ^  of  the  second,  and  i  of  the  thiid,  shall  be  e- 
qual  to  each  other. 

Prob.  49.  A  merchant  supported  himself  3  years  for  50 
pounds  a  year,  and  at  the  end  of  each  year,  added  to  that 
part  of  his  stock  which  was'  not  thus  expended,  a  sum 
equal  to  one  third  of  this  part.  At  the  end  of  the  third 
year,  his  original  stock  was  doubled.     What  was  that  stock  ? 

Ans.  740  pounds. 

Prob.  50.  A  general  having  lost  a  battle,  found  that  he 
had  only  half  of  his  army,  4*3600  men,  left  fit  for  action  ; 
i  of  the  army  -f600  being  wounded;  and  the  rest,  who 
were  ^  of  the  whole,  either  slain,  taken  prisoners,  or  missing. 
Of  how  many  men  did  his  army  consist  ?         Ans.  24000. 

For  the  solution  of  many  algebraic  problems,  an  acquain- 
tance with  the  calculations  of  powers  and  radical  quantities 
is  required.  It  will  therefore  be  necessary  to  attend  to 
these,  before  finishing  the  subject  of  equations* 


•<s^®^»" 


SECTION  VIIL 


INVOLUTION  AND  POWERS. 

Art.  198.   TT  HEN  a  quantity  is  multiplied  into  it- 
self, THE  PRODUCT  IS  CALLED  A  POWER. 

Thus  2x2=4,  tbe  square  or  second  power  of  2. 

2x2x2  =  8,  the  cube  or  third  power. 
2 X  2  X  2  X  2=16,  the  fourth  power,  &c. 
So  10  X  10=100,       the  second  power  of  10. 

10x10x10=1 000,      the  third  power. 
10x10x10x10  =  1 0000,  the  fourth  power,  &c. 


S4  ALGEBRA. 

And  aXa=^aa,       the  second  power  of  cf. 

ax  ax a—aaa,     the  third  power. 
ax  ax  ax  a=aaaa,  the  fourth  power,  &;c. 

199.  The  original  quantity  itself,  though  not,  like  the 
powers  proceeding  from  it,  produced  hy  multiplication,  is 
nevertheless  called  \he  first  power.  It  is  also  called  the  root 
of  the  other  powers,  because  it  is  that  from  which  they  are 
all  derived. 

200.  As  it  is  inconvenient,  especially  in  the  case  of  high 
powers,  to  write  down  all  the  letters  or  factors  of  which  the 
powers  are  composed,  an  abridged  method  of  notation  is 
generally  adopted.  The  root  is  written  only  once  ;  and  then 
a  number  or  letter  is  placed  at  the  right  hand,  and  a  little 
elevated,  to  signify  how  many  times  the  root  is  employed  as 
a  factor,  to  produce  the  power.  This  number  or  letter  is 
called  the  index  or  exponent  of  the  power.  Thus  a^  is  put 
for  ax  a  or  aa,  because  the  root  a  is  twice  repeated  as  a  fac- 
tor, to  produce  the  power  aa,  And  a^  stands  for  aaa ;  for 
here  a  is  repeated  three  times  as  a  factor. 

The  index  of  the^rs^  power  is  1  ;  but  this  is  commonly 
omitted.     Thus  a^  is  the  same  as  a. 

201.  Exponents  must  not  be  confoimded  with  co-efficients, 
A  co-efficient  shows  how  often  a  quantity  is  taken  as  depart 
of  a  whole.  An  exponent  shows  how  often  a  quantity  is 
taken  as  a   factor  in  a  product. 

Thus  ia=:a-\-a-\-a-{-a.  But  a«  =axaXaXfl. 

202.  The  scheme  of  notation  by  exponents  has  the  pe- 
culiar advantage  of  enabling  us  to  express  an  unknown  pow- 
er. For  this  purpose  the  index  is  a  letter,  instead  of  a  nu- 
merical figure.  In  the  solution  of  a  problem,  a  quantity 
may  occur,  which  we  know  to  be  some  power  of  another 
quantity.  But  it  may  not  be  yet  ascertained  whether  it  is 
a  square,  a  cube,  or  some  higher  power.  Thus  in  the  ex- 
pression a^,  the  index  x  denotes  that  a  is  involved  to  some 
power,  though  it  does  not  determine  what  power.  So  6*" 
and  J"  are  powers  of  h  and  d  ;  and  are  read  the  mth  power 
of  6,  and  the  ?ith  power  of  d.  When  the  value  of  the  in- 
dex is  found,  a  number  is  generally  substituted  for  the  letter. 
Thus,  if  m  =  3,  then  6"»=63  5  hut  if  m=5,  then  ^*"*=6^ 

203.  The  method  of  expressing  powers  by  exponents  is 
also  of  great  advantage  in  the  case  of  compound  quantities. 


POWERS.  85 

Thus«+6+rf|'ora+6+rf'  or   {a+b-{-dy,  is  (a-\-b+d)x 
(a+M-<^)x(a4-64-f^)  that  is,  thecube  of  (a+5+J).     But 
this  involved  at  length  vi^ould  be 
a^+3an-\'3a''d+3ab^-\-6oM+3ad^+b^-\-3b2d-^3hd^+dK 

204.  If  we  take  a  series*  of  powers  whose  indices  increase 
or  decrease  by  1,  we  shall  find  that  the  powers  themselves 
increase  by  a  common  midtiplkr,  or  decrease  by  a  common 
divisor ;  and  that  this  multiplier  or  divisor  is  the  original 
quantity  from  which  the  powers  are  raised. 

Thus  in  the  series    aaaaa^      aaaa,      aaa,      aa,       a ; 
Or  a^  a*  a^         a^       a'  ; 

the  indices  counted  from  right  to  left  are  1 ,  2,  3,  4,  5  ;  and 
the  common  difference  between  them  is  a  unit.  If  we  begin 
on  the  right,  and  multiply  by  a,  we  produce  the  several  pow- 
ers, in  succession,  from  right  to  left. 

Thus  axa=^a^  the  2d  term.  And  a^Xa^a"^. 

a^  X  «=a^  the  3d  term.  a*  xa=a^,  &c. 

If  we  begin  on  the  left,  and  divide  by  a, 
We  have  a^ —  «=«*.  And  a^-^«=a2, 

205.  But  this  division  may  be  carried  still  farther ;  and 
we  shall  then  obtain  a  new  set  of  quantities. 

a  11 

Thusa-7-«=--  =  l.  (Art.  128.)    And— 4- a  =  — .(Art.  163.) 
ct  di  act/ 

a  aa  aaa ' 

The  whole  series  then 

Is  aaaaa,  aaaa,  aaa,  aa,  a,  1,  — ,  — ,     ,  lyc. 

1       1       1      „ 

Or    a',    a%   a%    a%    a,    1,—,  —    — ,  Sic. 

Here  the  quantities  on  the  right  of  1,  are  the  reciprocals 
of  those  on  the  left.  (Art.  49.)  The  former,  therefore,  may 
be  properly  called  reciprocal  powers  of  a  ;  while  the  latter 
may  be  termed,  for  distinction  sake,  direct  powers  of  a.  It 
may  be  added,  that  the  powers  on  the  left  are  also  the  re- 
ciprocals of  those  on  the  right. 

*  Note.  The  term  series  is  applied  to  a  number  of  quantities  succeeding 
each  other,  in  some  regular  order.  It  is  not  confined  to  any  particular  law 
of  increase  or  decrease. 


86  ALGEBRA. 

For  1-f--— =lX-7-=«.  (Art.  162.)    .And  1-i-— =a^ 
la-  1 

206.  The  same  plan  of  notation  is  applicable  to  com- 
pound quantities.     Thus  from  a +6,  we  have  the  series, 

(a+by,  (a+by,  (a+b),  l,-^^,  j^^^^,  ^^„  <$,c. 

207.  For  the  convenience  of  calculation,  another  form  of 
notation  is  given  to  reciprocal  powers. 

11-,  11 

According  to  this,  "7"<^rrr=  a    .    And  ^«or— =a~^. 

11-2  11 

And  to  make  the  indices  a  complete  series,  with  1  for  the 

a 
common  diflference,  the  term-—  or  1,  which  is  considered  as 

no  power,  is  written  a**. 

The  powers  both  direct  and  reciprocal*  then, 

a      1       1       1         1 

Instead  of  ««««,«««,  aa,  a,    -,  -,  -,  --,  ^^,  &c. 

Will  be         «%   a%  aS  as   a%a-Stt-Sa"-%  a"*,  &c. 

Or              «+%  a+%a-^2^^+i^  ^o^  «-»,«-%  or'',  ar\  &c. 
And  the  indices  taken  by  themselves  will  be, 

+  4,  +3,  +2,  +1,  0,  -1,  -2,   -3,  -4,  Sic. 

208.  The  root  of  a  pow  er  may  be  expressed  by  more  let- 
ters than  one. 

Thus  aa  X  aa,  or  aa|^  is  the  second  power  of  aa. 

And  aaxaaxaa,  or  aa\    is  the  third  power  of  aa,  &c. 

Hence  a  certain  power  of  one  quantity,  may  be  a  differ- 
ent power  of  another  quantity.  Thus  «*  is  the  second  pow- 
er of  a^y  and  the  fourth  power  of  a. 

209.  All  the  powers  of  1  are  the  same.  For  1x1,  or 
1x1x1,  &:c.  is  still  I. 

*  See  Note  E. 


INVOLUTION.  87 


INVOLUTION. 


210.  Involution  is  finding  any  power  of  a  quantity,  by 
multiplying  it  into  itself.  The  reason  of  the  following  gen- 
eral rule  is  manifest,  from  the  nature  of  powers. 

Multiply  the  quantity  into  itself,  till  it  is  taken 
as  a  factor,  as  many  times  as  there  are  units  in  the  in- 
dex of  the  power  to  which  the  quantity  is  to  be  rais- 
ED. 

This  rule  comprehends  all  the  instances  which  can  occur 
in  involution.  But  it  will  be  proper  to  give  an  explanation 
of  the  manner  in  which  it  is  applied  to  particular  cases. 

211.  A  single  letter  is  involved,  by  giving  it  the  index  of 
the  proposed  power ;  or  by  repeating  it  as  many  times,  as 
there  are  units  in  that  index. 

The  4th  power  of  a,  is  a*  or  aaaa.     (Art.  198.) 

The  6th  power  of  y,  is  y^  ^^  yyyyyy* 

The  nth  power  of  a;,  is  x'^  or  xxx  . . .  n  times  repeated- 

212.  The  method  of  involving  a  quantity  which  consists  of 
several  factors,  depends  on  the  principle,  that  the  power  of 
the  product  of  several  factors  is  equal  to  the  product  of  their 
powers. 

Thus  {ayY  =  a^y^ ,     For  by  art.  210  ;  (ayY  =ay  X  ay. 
But  ayxay=ayay=aayy=a^y^. 

So  (hmxy  =  hmxxhmxxbmx—hhhrnrnrnxx30^h^rn^x^. 
And  (ady)»=sadyxadyxady  . . .  n  times  =a"£?y^ 

In  finding  the  power  of  a  product,  therefore,  we  may  ei- 
ther involve  the  whole  at  once  ;  or  we  may  involve  each  of 
the  factors  separately,  and  then  multiply  their  several  pow- 
ers into  each  other. 

Ex.  1.  The  4tli  power  of  dhy,  is  [dhyY,  or  d^h*y*. 

2.  The  3d  power  of  4b,  is  (46)%  or A^b\  or  646^ 

3.  The  nib  power  of  Sad,  is  (6«J)%  or  6*a"cZ". 

4.  The3dpowerof3mx2y,is(3mX2y)%or27m3X8:y'« 

213.  A  compound  quantity,  consisting  of  terms  connec- 
ted by  -f  and  — ,  is  involved^by  an  actual  multiplication  of 
it^  several  parts.    Thus, 


88  ALGEBRA 

(a+6)'=ia+6,  the  first  power. 
a^+  ah 


(a-|-6)2=a2+2a6-f6%  the  second  power  of  (a -j- 6). 
a  +6 


(a+6)^=a=*  +  3a26  +  3a62+63,  the  third  power. 
a  -{■  b 


«4  ^3^3  ^_l_  3^252  ^^^,3 


(a4.5)*=a*+4a35+6a2 62 ^4^53^2,4^  the  4th  power,  kc. 

2.  The  square  of  «— 5,  is  a^  _  2a6+Z>2 . 

3.  The  cube  of  a+l,  is  a^ +3«2  4.3«+l. 

4.  The  square  a-^-b  ^h,  isa^-\-2ab'{-2ah'{-b^+2bh+h'', 

5.  Required  the  cube  of  a-\-2d-{-3. 

6.  Required  the  4th  power  of  b-{-2, 
?•  Required  the  5th  power  of  a;  + 1 . 
8.  Required  the  6th  power  of  1—b* 

21 4.  The  squares  of  binomial  and  residual  quantities  oc- 
cur so  frequently  in  algebraic  processes,  that  it  is  important 
to  make  them  familiar. 

If  we  multiply  a+k  into  itself,  and  also  a—h, 

We  have  a+k  And  a— A 

a-i-k  a—k 


a^-^ah  a^  ^ah 

+ah-\-h''  -ah+h' 


a2  4-2aA+^^  a^-2ah+h^. 

Here  it  will  be  seen  that,  in  each  case,  the  first  and  last 
terms  are  squares  of  a  and  h  ;  and  that  the  middle  term  is 
twice  the  product  of  a  into  h»    Hence  ti^e  squares  of  bino- 


INVOLUTION.  89 

mial  and  residual  quantities,  without  multiplying  each  of  the 
terms  separately,  may  be  found,  by  the  following  proposi- 
tion.* 

The  square  Of  a  binomial,  the  terms  of  which  are 

BOTH  positive,  IS  EQUAL  TO  THE  SQUARE  OF  THE  FIRST  TERM, 
+  TWICE  THE  PRODUCT  OF  THE  TWO  TERMS  ;  +  THE  SQUARE 
OF  THE  LAST  TERM. 

And  the  square  of  a  residual  quantity,  is  equal  to  the  square 
of  the  first  term,  —  twice  the  product  of  the  two  terms,  + 
the  square  of  the  last  term. 

Ex.  1.  Thesquareof  2a+J,  is  4«2+4a&  +  &^ 

2.  The  square  of  h+1,  isA^-HSA+l. 

3.  The  square  of  ab+cd^  is  a^b^-^2abcd-^c^d^, 

4.  The  square  of  6^+3,  is  36i/^+ 36^+9. 

5.  The  square  of  3d—h,  is  9d^—6dh+h^» 

6.  The  square  of  a— 1,  is  a*— 2a+l. 

For  the  method  of  finding  the  higher  powers  of  binomi- 
als, see  one  of  the  succeeding  sections. 

215.  For  many  purposes,  it  will  be  sufficient  to  express 
the  powers  of  compound  quantities  by  exponents,  without  an 
actual  multiplication. 

Tlius  the  square  of  a+b,  is  a+6J%  or  (a'{'bf.     Art.  203. 
The  nth  power  of  6c +8  + a;,  is  (^c+S  +  x)". 

In  cases  of  this  kind,  the  vinculum  must  be  drawn  over  all 
the  terms  of  which  the  compound  quantity  consists. 

216.  But  if  the  root  consists  of  several  factors,  the  vincu- 
lum which  is  used  in  expressing  the  power,  may  either  ex- 
tend over  the  whole ;  or  may  be  applied  to  each  of  the  fac- 
tors separately,  as  convenience  may  require. 

Thus  the  square  of  a-f  ftxc-ff/,  is  either 

a-{-bxc+d\  or  a+bfxc-^d\^. 

For  the  first  of  these  expressions  is  the  square  of  the  pro- 
duct of  the  two  factors,  and  the  last  is  the  product  of  their 
squares.     But  one  of  these  is  equal  to  the  other.  (Art.  212.) 

The  cube  of  axb  +  d,  is  {axb-^d)\  or a^ X {b -{- d)\ 

'*  Euclid's  Elemeats,  Book  ii.  Prop.  4. 
13 


90  ALGEBRA. 

217.  Wheu  a  quantity  whose  power  has  been  expressed 
by  a  vinculum  and  an  index,  is  afterwards  involved  by  an  ac* 
tual  multiplication  of  the  terms,  it  is  said  to  be  expanded. 

Thus  (a+6)^  when  expanded,  becomes  d^+2ab+P. 
And  (a-h6-fA)S  becomes  a^  +^ab'{-2ah+b^  -{-^h-^hK 

218.  With  respect  to  the  sign  which  is  to  be  prefixed  to 
quantities  involved,  it  is  important  to  observe,  that  when  the 

ROOT  IS  POSITIVE,  ALL  ITS  POWERS  ARE  POSITIVE  ALSO  ;  BUT 
WHEN  THE  ROOT  IS  NEGATIVE,  THE  ODD  POWERS  ARE  NEGA- 
TIVE, WHILE  THE  EVEN  POWERS  ARE  POSITIVE. 

For  the  proof  of  this,  see  art.  1.09. 

The  2d  power   of  — «  is  4'«^ 
The  3d  power  is  —  a^ 

The  4th  power  is  +a* 

The  5th  power  is  — a^,  &c. 

219.  Hence  any  odd  power  has  the  same  sign  as  its  root. 
But  an  even  power  is  positive,  whether  its  root  is  positive  or 
negative. 

Thus  +ax  +a=a2 

And    —ax  —a==a^, 

220.  A  QUANTITY  WHICH  IS  ALREADY  A  POWER,  IS  INVOLV- 
ED BY  MULTIPLYING  ITS  INDEX,  INTO  THE  INDEX  OF  THE  POW- 
ER TO  WHICH  IT  IS  TO  BE   RAISED. 

1.  The  3d  power  of  a%  is  a^^^=za^. 

For  «2  =aa:  and  the  cube  of  aa  is  aaxaax  aa=:aaaaaa=za^ ; 
which  is  the  6th  power  of  a,  but  the  3d  power  of  a^. 
£     For  a  farther  illustration  of  this  rule,  see  arts.  233,  4. 

2.  The  4th  power  of  a^jsig  ^3X4^2X4  _^i2  58^ 

3.  The  3d  power  of  Aa^x,  is  64a^x^. 

4.  The  4th  power  of  ^a^'.xSx^d,  is  16«*^  x81a?«</% 

5.  The  5th  power  of  («+i)%  is  (a+by\ 

6.  The  nth  power  of  a^,  is  a^". 

7.  The  nth  power  of  (x— y)"*,  is  (a;— y)"'". 


8.  a^+b'\^=a^-h^a'b'+b^.     (Art.  214.) 

9.  ?x6^2=a«x6«.        10.  {a'b^h'')^=:an^h' 


INVOLUTION. 


9* 


221.  The  rule  is  equally  applicable  to  powers  whose  ex- 
ponents are  negative* 

Ex.  1.  The  3d  power  of  a"^,  is  a~2X3--^-6^ 
1 
Fora-2=— .     (Art.  207.)     And  the  3d  power  of  this  is 

L    JL    J-__J_ L_  -6 

aa     aa     aa"^  aaaaaa~a^'^ 

2.  The  4th  power  a^b"^,  is  a^b"^^  ^,  or  ttI* 

3.  The  cube  of  2a?«^-"»,  is  Sx^^^-^'". 

4.  The  square  of  b^x"^,  is  6®a;~*. 

5.  The  nth  power  of  x"""*  5  is  »"•""*,  or-i;^. 

222.  It  must  be  observed  here,  as  in  art.  218,  that  if  the 
sign  which  is  prejloeed  to  the  power  be  — ,  it  must  be  changed 
to  -f ,  whenever  the  index  becomes  an  even  number. 

Ex.  1.  The  square  of  —  a»,  is  ■\-a^ ,  For  the  square  of 
— a^,  is  — a^x-— a',  which,  according  to  the  rules  for  the 
signs  in  multiplication,  is  +a®. 

2.  Butthecwfteof -a^,is-a^  For-a' X -a'' X -«'=-«'. 

3.  The  square  of  —a;",  is  -f  a?^". 

4.  The  nth  power  of  — a^,  is  _a^". 

Here  the  power  will  be  positive  or  negative,  according  as 
the  number  which  n  represents  is  even  or  odd. 

223.  A  FRACTION  IS  INVOLVED,  BY  INVOLVING  BOTH  THE 
NUMERATOR,  AND  THE  DENOMINATOR. 

a     a^ 

1.  The  square  of  "^is^.     For,  by  the  rule  for  the  mul- 

plication  of  fractions,  (Art.  155.) 

a       a      aa     a^ 
T^T'^'bb^b^* 

1  11  1 

2.  The  2d,  3d,  and  nth  powers  of—,  are  "y,  -7  and  r^. 

a  a    u  a 

2xr^  8^3  y.6 

3.  The  cube  of  -^~,   is 


3y  '         27^ 


3  • 


4.  The  nth  power  of -^,  is  -^^. 


92  ALGEBRA. 

5.  Tlie  square  of (^JJi ,  is  — f^^^^y — • 

6.  The  cube  of -^:r^,  is  — ^?.     (Art.  221.) 

224.  Examples  of  binomials^  in  which  one  of  the  terms  is 
a  fraction. 

1.  Find  the  square  of  ^+1,  and  a:— ^,  as  in  art.  214. 


2  4a      4 

2.  The  square  of  «+— ,  is  a^  ^-— _|.— ^ 

3.  The  square  of  oc+-^^  is  x^  -\-hx-\--T* 

h  ^hx     h^ 

4.  Thesquareof  a:-— ,  is  x*  -"^+m2' 


225.  It  has  been  shown,  (Art.  165,)  that  a  fractional  co- 
efficient may  be  transferred,  from  the  numerator  to  the  de- 
nominator of  a  fraction,  or  from  the  denominator  to  the  nu- 
merator. By  recurring  to  the  scheme  of  notation  for  recip- 
rocal powers,  (Art.  207,)  it  will  be  seen  that  any  factor 
may  also  be  transferred,  if  the  sign  of  its  index  be  changed. 

ax"^ 

1.  Thus,  in  the  fraction—     ,  we  may  transfer  x  from  the 

if 

numerator  to  the  denominator, 

wxr*      a        ^       a       \        a 
For-     =yxa:  ^^-x^=^. 

u 

2.  In  the  fraction  7-3 ,  we  may  trausfer  y  from  the  de- 
nominator to  the  numerator. 


a        a       \       a  ay"^ 

~bf^  =T  ^y^  ~T  X5^^='6~- 


INVOLUTION. 

dar'       d 
•     x^  ""x^^a** 

h        by-'' 
4-  ^^^=-a- 

226.  In  the  same  manner,  we  may  transfer  a  factor  which 
has  a  positive  index  in  the  numerator,  or  a  negative  index 
in  the  denominator. 

ax^         a 

1.  Thus-7-  =  T-z:s.     For  x^  is   the   reciprocal  of  a:~^, 

1  €IX^  CI 

(Arts.  205,  207,)  that  is,  x^  =~~^'     Therefore  -^=-^^. 

h        hy^  ad^      ay^ 

2.   =  -^-  3 =— ^^ — 

by^       b   '  '    xy"^      xd'~^' 

227.  Hence,  the  denominator  of  any  fraction  may  be  en- 
tirely removed,  or  the  numerator  may  be  reduced  to  a  unit, 
without  altering  the  value  of  the  expression. 

a         1 

1.  Thus  T"=T7^j  or  a6"~^ 

x-^  1 

a?^a"""*  1 


ADDITION  AND  SUBTRACTION  OF  POWERS. 

228.  It  is  obvious  that  powers  may  be  added,  like  other 
quantities,  by  writing  them  one  after  another,  with  their  signs 
(Art.  69.) 

Thus  the  sum  of  a^  and  6^,  is  a'  +6^. 

And  the  sum  of  a*  -6"  and  h'  -d*,  is  a^  -b^'+h'  -d*. 

229.  The  same  powers  of  the  same  letters  are  like  quanti- 
ties ;  (Art.  45,)  and  their  co-efficients  may  be  added  or  sub 
tracted,  as  in  arts.  72  and  74. 

Thus  the  sum  of  2a *  and  3a*,  is  5a ^. 

It  is  as  evident  that  twice  the  square  of  a,  and  three  times 
the  square  of  a,  are  five  times  the  square  of  a,  as  that  twice 
a  and  three  times  a,  are  five  times  c. 


<)4  ALGEBRA. 

To  —3x^1/^  3S'"  3ay       -ba^h^  3(«+3/y' 

Add— 2a;«^*  66"*       —la'^xf  ea^h^  4(«+^)« 

230.  But  powers  of  different  letters,  and  different  powers 
of  the  ^orne  letter,  must  be  added  by  writing  them  down  with 
their  signs. 

The  sum  of  a^  and  a^,  is  a^  +a^. 

It  is  evident  that  the  square  of  a,  and  the  cube  of  a,  are 
neither  twice  the  square  of  a,  nor  twice  the  cube  of  a. 
The  sum  of  aH""  and  3a^b^,  is  a^fe"  +3a*6«. 

231.  Subtraction  of  powers  is  to  be  performed  in  the  same 
manner  as  addition,  except  that  the  signs  of  the  subtrahend 
are  to  be  changed  according  to  art.  82. 
From  2a*  -36^  Sh'^b^  a^b""  5(a-i 
SuU.      -Q>a''               4b''           ^hH^         an^"         21 


Diff.     i  Sa*  -yi^      -^hH^     ''^•'^•"^        3{a^hy 


MULTIPLICATION  OF  POWERS. 

232.  Powers  may  be  multiphed,  like  other  quantities,  by 
writing  the  factors  one  after  another,  either  with,  or  without, 
the  sign  of  multipHcation  between  them.    (Art.  93.) 

Thus  the  product  of  a^  into  6^,  is  a ^6^,  or  aaabb. 
Mult.   x~^        h^b-"        3ay  dh^x"''       a^'b^y^ 

Into      ft"'         «^  ~2x  Aby'^  a'^b'^y 

Prod.  «'"a:~2'^''<^^  '-    — 6a«a:t^2     ^.^^-\<f.k^x- '^  a^b'^y^a^'b^y 


The  product  in  the  last  example,  may  be  abridged,  by 
bringing  together  the  letters  which  are  repeated. 
It  will  then  become  a^¥y^. 

The  reason  of  this  will  be  evident,  by  recurring  to  the 
series  of  powers  in  art.  207,  viz. 
(7/fS     G•^3,     a+%     a+S     a°,     a-^     a"^,     or^,     oT*,  kc. 
Or,  which  is  the  same, 

11  1  1       , 

aaaa,     aaa,    aa,    a,         I,     -,   -,      -^,     ««««' ^^ 


POWERS.  -9^ 

By  comparing  the  several  terms  with  each  other,  it  will 
be  seen  that  if  any  two  or  more  of  them  be  multiplied  to- 
gether, their  product  will  be  a  power  whose  exponent  is  the 
sum  of  the  exponents  of  the  factors. 

Thus  a-  xa^  =aaxaaa=aaaaa—a^ , 

Here  5,  the  exponent  of  the  product,  is  equal  to  2+3,  the 
sum  of  the  exponents  of  the  factors. 

Soa"X«'"=a"'^"*. 

For  a",  is  a  taken  for  a  factor  as  many  times,  as  there  are 
units  in  n ; 

And  a*",  is  a  taken  for  a  factor  as  many  times,  as  there  are 
units  in  m  ; 

Therefore  the  product  must  be  a  taken  for  a  factor  as 
many  times  as  there  are  units  in  both  m  and  n.     Hence, 

233.  Powers  of  the  same  root  may  be  multiplied,  by 

ADDING    their    EXPONENTS. 

Thus  a"  X  a«  =a'''^^=a\  And  x^Xx^x  Ar=a;=^+2+i  -_,p6 ^ 
Mult,     ^ar-         3a:*         523/3  a'b^y^  (fi  +  h-yy' 

Into       2a"         2a:;3  j4^  ^3^2^  h-^-h-y 

Prod.    8a2«      ^^'        h'y^         ^SHJ^  (j^.^.,^ 

Mult.  x^-{ x^y-{-xy^  +y^  into  x—y.        Ans.  a?*—?/*. 
Mult.  4^^i/-l-3a?y— 1  into  2x^^x. 
Mult.  x^-{-x-5into2x^'\-x+l. 

234.  The  rule  is  equally  applicable  to  powers  whose  ex- 
ponents are  negative, 

1.  Thus  a-^  X  a-^=a-'.       That  is  ~  x = 

aa     aaa     aaaaa 

2.  2/-«  x^-"=r"*"      That  is  ^  X^=^. 

3.  -a-2xa~^=-a-^  4.  a-2xa^=a^"-^=a^ 


5.  «-«  x  a'"=o'"-«.  6.  3/-2  x/  =2/ 


235.  If  «-{-&  be  multiplied  into  «  — 5,  the  product  will  be 
a'-lr^:  (Art.  110.)  that  is, 


9c  algebra. 

The  product  of  the  sum  and  difference  of  two  quan- 
tities, IS  EQUAL  TO  THE  DIFFERENCE  OF  THEIR  SQUARES. 

This  is  another  instance  of  the  facihty  with  which  gene- 
ral  truths  are  demonstrated  in  algebra.  See  arts.  23  and 
77. 

If  the  sum  and  difference  of  the  squares  be  multiphed, 
the  product  will  be  equal  to  the  difference  of  the  fourth 
powers,  &;c. 

Thus  (a-y)  x  {a-{-y)=a^-y\ 

(a*  -y")  X  {a'  +^')=a'  -y\  &c. 

DIVISION  OF  POWERS. 

236.  Powers  may  be  divided,  like  other  quantities,  by  re- 
jecting from  the  dividend  a  factor  equal  to  the  divisor ;  or 
by  placing  the  divisor  under  the  dividend,  in  the  form  of  a 
fraction. 

Thus  the  quotient  oi  a%^  divided  by  h\  is  a^.  (Art.  116.) 
Divide  9ay  I26V  a^b-ir^ay  dx{a-h-\-yf 
By  -3a^  26^  a"  {a-h-\-yf 

Quot.        —3/  b+3y*  d 


The  quotient  of  a^  divided  by  a ^,  is  ~.  But  this  is  equal 
to  a^.     For,  in  the  series 

a+^  a+^  «+^  a+^  a°,  a^^  a'^,  a^^  ar\  &c. 

if  any  term  be  divided  by  another,  the  index  of  the  quotient 
will  be  equal  to  the  difference  between  the  index  of  the  divi- 
dend, and  that  of  the  divisor. 

„     aaaaa  «*" 

Thus  a'-^a^^ ^a^.     And  «^»-^a^=-^=;a*«~\ 

aaa  a 

Hence, 

237.  A  POWER  MAY  BE  DIVIDED  BV  ANOTHER  POWER  OF 
rFHE  SAME  ROOT,  BY  EXTRACTING  THE  INDEX  OF  THE  DIVISOF; 
PROM  THAT  OF  THE  DIVIDEND. 


POWERS.  97 

Thus  y^-^y^  =^y^^  =t^i.    That  is-^  =y. 

And  a«+i^a=a'»+i-"i=a".     That  is  -^=«^ 

if'' 
Anda;'»-T-a;"=a;'^-"=a;°=l.  Thatis:3;;i=l. 

Divide   /"^  6«  8a"+^  a"+'  12(5+^)" 

By         ^'^  63  4a"'  a*  3(6+y)3 

Quot.      3/2  '>^  20^  4{b+yY'-^ 

238.  The  rule  is  equally  applicable  to  powers  whose  ex- 
ponents are  negative. 

I.  The  quotient  of  a"**  by  a~^,  is  a""^. 

1  1  1  a«a       «aa        1 

That  is ~  — --== X  -r-  = 


aaaaa  '  aaa      aaaaa        1       aaaaa     aa 

1  1        aj3  1 

^-5^  ^-3^  _ ^-2^     That  is  3^ 


,       1  A 

3.  A»-7-A-^=A'+^=A3.       Thatis^*-^^=^«XY=^^ 

4.  6a"-4-2a~^=3a"+3.  5.  ha'^-^a^ba^. 

6.    ^3^65  =2/3-5 -,J~2,  7.    ^4^  ^7 -.^-3^ 

9.  (J  +  aj)"-^(6  +  a;)  =  (^^+^)''"'. 

The  multipHcation  and  division  of  powers  by  adding  and 
subtracting  their  indices,  should  be  made  very  familiar ;  as 
they  have  numerous  and  important  apphcations,  in  the  high- 
er branches  of  algebra. 

EXAMPLES  OF  FRACTIONS  CONTAINING  POWERS. 

239,  In  the  section  on  fractions,  the  following  examples 
were  omitted,  for  the  sake  of  avoiding  an  anticipation  of 
the  subject  of  powers. 

5a*  5a» 

1.  Reduce  — -  to  lower  terms.     Ans.  ~r~. 

5a*      5aaaa      baa 

Qx^  2a? 

2.  Reduce  -^  to  lower  terms.      Aos.  y  or  2a?- 

14 


4.  Reduce  — '  n  2   _l!i^„2 — "~~  to  lower  terms. 


98  ALGEBRA. 

3fl5*-f-4a6                                  .       3tf+4a3 
3.  Reduce  — —2 —  t^  lower  terms,    Ans.  — ^ . 

Aa^—Gay+Sy"  ,     ^ 

Ans. g    ,  2 — —obtained  by  dividing  each  term  by  2ay. 

a  a~^ 

5.  Reduce  -^  and  -^r?  to  a  common  denominator. 

a^  Xa"-*  is  a""^,  the  first  numerator.     (Art.  146.) 
a  3  xa"~^  is  a°  =1,  the  second  numerator. 
a^X  a~*  is  a"^,  the  common  denominator. 

The  fractions  reduced  are  therefore  — i  and  — n. 


2a^         «2 

6.  Reduce  r-j  and  — ,  to  a  common  denommator. 

2«8  5a^        2a»  5 

Ans,~  and  5^,  or--^  and  -^     (Art.  145.) 

3a;2  Ja;  Sc^a;^        3(? 

7.  Multiply  -^3  into  -^ .     Ans.  -^  =^^. 

8.  Multiply—^-,  into — ^.     *   '  " 

of  4-1  6^-1 

9.  Multiply— ^-,mto-^:jp^. 

10.  Multiply— ^,  into—  ,  and —3. 

11.  Divide^^by^.     Ans.^p=^. 


«•*— a; 


12.  Divide  — -3 — ,  by 


a'     '    ->  a 


13.  Divide— ^  by— p- 

14.  Divide  ~^~M  T'' 


SECTION  IX. 


EVOLUTION  AND  RADICAL  QUANTITIES.* 

Art.  240.  xF  a  quantity  is  multiplied  into  itself,  the  pro- 
duct is  a  power.  On  the  contrary,  if  a  quantity  is  resolved 
into  any  number  of  equal  factors^  each  of  these  is  a  root  of 
that  quantity. 

Thus  J  is  a  root  of  hhh  ;  because  hhh  may  be  resolved  in- 
to the  three  equal  factors  Z>,  and  6,  and  h. 

In  subtraction,  a  quantity  is  resolved  into  two  parts. 

In  division,  a  quantity  is  resolved  into  two  factors. 

In  evolution,  a  quantity  is  resolved  into  equal  factors, 

241.  A  ROOT  OF  A  QUANTITY,  THEN,  IS  A  FACTOR,  WHICH 
MULTIPLIED  INTO  ITSELF  A  CERTAIN  NUMBER  OF  TIMES,  WILL 
PRODUCE  THAT  QUANTITY. 

The  number  of  times  the  root  must  be  taken  as  a  factor, 

to  produce  the  given  quantity,  is  denoted  by  the  name  of 

the  root. 

Thus  2  is  the  4th  root  of  16  ;  because  2x2x2x2  =  16, 

where  2  is  taken  four  times  as  a  factor,  to  produce  IQ. 
So«3  is  the  square  root  of  «^  ;  for  a^  xa^  =««.  (Art.  233.) 
And  a^  is  the  cube  root  of  a^  ;   fora^  xa^  xa^=a^. 
And  a  is  the  6th  root  of  a®  5  for  ax  ax  ax  ax  ax  a=a^. 
Powers  and  roots  are  correlative  terms.     If  one  quantity 

is  a  power  of  another,  the  latter  is  a  root  of  the  former. 

As  6^  is  the  cube  of  6  ;  6  is  the  cube  root  of  6^. 

242.  There  are  two  methods  in  use,  for  expressing  the 
roots  of  quantities,  one  by  means  of  the  radical  sign  v',  and 
the  other  by  a  fractional  index.  The  latter  is  generally  to 
be  preferred.  But  the  former  has  its  uses  on  particular  oc- 
casions. 

*  Newton's  Arithmetic,  Maclaurin,  Emerson,  Euler,  Saunderaoii,  and 
Simpson. 


100  ALGEBRA. 

When  a  root  is  expressed  by  the  radical  sign,  liie  sign 
is  placed  over  the  given  quantity,  in  this  manner,  Va. 
Thus  ^y/a  is  the  2d  or  square  root  of  a, 

^  V^  is  the  3d  or  cube  root. 

V«  is  the  nth  root. 
And      "\/a+^  is  the  nth  root  of  a+y. 

243.  The  figure  placed  over  the  radical  sign,  denotes  the 
number  of  factors  into  which  the  given  quantity  is  resolved ; 
in  other  w^ords,  the  number  of  times  the  root  must  be  taken 
as  a  factor,  to  produce  the  given  quantity. 

So  that  V«x2Vrt=a. 

And       *v/«X  ^V«X  'v^fl^=«. 

And        V«X  V«  •  .  •  •  w  times  =«. 
The  figure  for  the  square  root  is  commonly  omitted  ;  yfa 
being  put  for  ^y/a.     Whenever,  therefore,  the  radical  sign 
is  used  without  a  figure,  the  square  root  is  to  be  understood. 

244.  When  a  figure  or  letter  is  prefixed  to  the  radical  sign, 
without  any  character  between  them  ;  the  two  quantities  are 
to  be  considered  as  multiplied  together. 

Thus  2v/«,  is  2  X  \/a,  that  is,  2  multiplied  into  the  root  of 
a,  or,  which  is  the  same  thing,  twice  the  root  of  a. 

And  x^Jh,  is  a:  X  V^j  or  x  times  the  root  of  5. 

When  no  co-efficient  is  prefixed  to  the  radical  sign,  1  is 
always  to  be  understood  ;  yja  being  the  same  as  1  /ti,  that 
is,  once  the  root  of  «. 

245.  The  method  of  expressing  roots  by  radical  signs, 
has  no  very  apparent  connection  with  the  other  parts  of  the 
scheme  of  algebraic  notation.  But  the  plan  of  indicating 
them  hy  fractional  indices^  is  derived  directly  from  the  mode 
of  expressing  powers  by  integral  indices.  To  explain  this, 
let  «^  be  a  given  quantity.  If  the  index  be  divided  into  any 
number  of  equal  parts,  each  of  these  will  be  the  index  of  a 
root  of  a^. 

Thus  the  square  root  of  a^,  is  «^.  For,  according  to  the 
definition,  (Art.  241,)  the  square  root  of  a«  is  a  factor,  which 
multiplied  into  itself  will  produce  a^  But  a^xa^=a^. 
(Art.  233.)  Therefore,  a^  is  the  square  root  of  a«.  The 
index  of  the  given  quantity  a®,  is  here  divided  into  the  two 
equal  parts  3  and  3.  Of  course,  the  quantity  itself  is  resolv- 
ed into  the  two  equal  factors  a^  and  a^. 


RADICAL  QUANTITIES.  Iqi 

The  cube  root  of  a^  is  a^.     For  a^  Xa^  xa^=a^. 

Here  the  index  is  divided  into  three  equal  parts,  and  the 
quantity  itself  resolved  into  three  equal  factors. 

The  square  root  of  a^  is  a^  or  a.     For  axa=a^. 

By  extending  the  same  plan  of  notation,  fractional  indices 
are  obtained. 

Thus,  in  taking  the  square  root  of  a  *  or  «,  the  index  1  is 
divided  into  the  two  equal  parts  i  and  i ;  and  the  root  is  d^. 
On  the  same  principle. 

The  cube  root  of  a,  is  a^  =  ^y/a. 

x 
The  wth  root,  is  a"  =  "/a,  kc. 


And  the  nth  root  of  a -fa;,  is  (r'  +  a:)n  =Va-f  a;. 

246.  In  all  these  cases,  the  denominator  of  the  fractional 
index,  expresses  the  number  of  factors  into  which  the  given 
quantity  is  resolved. 

ILL  1  L  > 

So  that  a^  Xa^  X  a^  =«.    And  a"  X  a" ....  n  times  =a, 

247.  It  follows  from  this  plan  of  notation,  that 


J.  X  ±4-1.  JL_L.' 

a=*  X a^  =a2 ^\     Fora^^^  =a^ 


or  a* 


a^xa^xa^=a3^^^=*=a%  (Sic. 

where  the  multiplication  is  performed  in  the  same  manner,  as 
the  multiplication  of  powers,  (Art.  233,)  that  is,  by  adding 
the  indices, 

248.  Every  root  as  well  as  every  power  of  1  is  1.  (Art. 
209.)  For  a  root  is  a  factor  which  multiplied  into  itself  will 
produce  the  given  quantity.  But  no  factor  except  1  can  pro- 
duce 1 ,  by  being  multiplied  into  itself. 

So  that  1" ,  1,  v'l,  Vlj  <Szc.  are  all  equal. 

249.  Negative  indices  are  used  in  the  notation  of  roots,  as 
well  as  of  powers.     See  art.  207. 

1  , 

Thus  —=«""¥ 


—  -a— 5 

1        _. 

a^ 

a« 

102  ALGEBRA. 


POWERS  OF  ROOTS. 

250.  It  has  been  shown  in  what  manner  any  power  or  root 
may  be  expressed  by  means  of  an  index.  The  index  of  a 
power  is  a  whole  number.  That  of  a  root  is  a  fraction  whose 
numerator  is  1.  There  is  also  another  class  of  quaiitities, 
which  may  be  considered,  either  as  powers  of  roots,  or  roots 
of  powers. 

Suppose  a^  is  multiplied  into  itself,  so  as  to  be  repeated 
three  times  as  a  factor. 

The  product  a*"^^"^*  or  a^  (Art.  247,)  is  evidently  the 

cube  of  a",  that  is,  the  cube  of  the  square  root  of  a.  This 
fractional  index  denotes,  therefore,  a  power  of  a  root.  The 
denominator  expresses  tlte  root,  and  the  numerator  the  pow- 
er. The  denominator  shows  into  how  many  equal  factors  or 
roots  the  given  quantity  is  resolved;  and  the  numerator 
shows  how  many  of  these  roots  are  to  be  multiplied  to- 
gether. 

Thus  a^  is  the  4th  power  of  the  cube  root  of  a* 

The  denominator  shows  that  a  is  resolved  into  the  three 

J  JL  i 

liictors  or  roots  a^,  and  a^ ,  and  a'^.  And  the  numerator 
shows  that  four  of  these  are  to  be  multiplied  together ;  which 

will  produce  the  fourth  power  of  a^  ;  that  is, 

i  i  A.  J.  4. 

^51.  As  a^  is  a  power  of  a  root,  so  it  is  a  root  of  a  power. 

Let  a  be  raised  to  the  third  power  a^.     The  square  root  of 

"^       1 
this  h  a^.     For  the  root  of  a^  is  a  quantity  which  multipUed 

into  itself  will  produce  a^. 

2.         1         1         ^- 

But  according  to  art.  247,  a^=a^Xa^xa^  j  and  this 
multiplied  into  itself,  (Art.  1 03,)  is 

XL         J.         J.         1         1 

a^  Xa^  X  a'^\  a^  Xa^xa'^=a^. 
Therefore  a*  is  the  square  root  of  the  cube  of  cu 

m 

In  the  same  manner,  it  may  be  shown  that  o"  is  the  mih 
power  of  the  nth  root  of  a  ;  or  tlie  nth  root  of  the  mth  pow- 


RADICAL  QUANTITIES.  203 

er :  that  is,  a  Toot  of  a  power  is  equal  to  the  same  power  of 
the  same  root.  For  instance,  the  fourth  power  of  the  cube 
root  of  a,  is  the  same,  as  the  cube  root  of  the  fourth  pow- 
er of  a. 

252.  Roots,  as  well  as  powers,  of  the  same  letter,  may  be 
multiplied  by  adding  their  exponents,  (Art.  247.)  It  will  be 
easy  to  see,  that  the  same  principle  may  be  extended  to 
powers  of  roots,  when  the  exponents  have  a  common  de- 
nominator. 

Thus  a^xa"^=a^^^=a\ 

For  the  first  numerator  shows  how  often  a '  is  taken  as  a 

2 

factor  to  produce  a'' .     (Art.  250. J 

And  the  second  numerator  shows  how  often  a^  is  talien  as 

3 

a  factor  to  produce  a^. 

The  sum  of  the  numerators,  therefore,  shows  how  often 
the  root  must  be  taken,  for  the  product*    (Art.  103.) 

2x1. 

Or  thus,  a'^=a'^  Xa\ 
And  a^=a'^  Xa'^  xa^» 

2  311  1115 

Therefore  a^ xa^ =a^ xa^ Xa^ Xa^ xa ^ —a^ . 

253.  The  value  of  a  quantity  is  not  altered,  by  applying 
to  it  a  fractional  index  whose  numerator  and  denominator 
are  equal. 

^  3        n^ 

Thus  «=rt'^=a^=rtn  For  the  denominator  shows  that 
a  is  resolved  into  a  certain  number  of  factors  ;  and  the  nu- 

merator  shows  that  all  these  factors  are  included  in  a"  * 

3  1  J.  A 

Thus  a^  =a^  x  a^  X  a^,  which  is  equal  to  «. 

n       X        i        i 
And   a»=za^  xa"^  xa''  .  . . .  n  times. 

On  the  other  hand,  when  the  numerator  of  a  fractional 
index  becomes  equal  to  the  denominator,  the  expression 
may  be  rendered  more  simple  by  rejecting  the  index. 

n 

Instead  of  ««",  we  may  write  a, 

254.  The  index  of  a  power  or  root  may  be  exchanged, 
for  any  other  index  of  the  same  value. 

Instead  of  a^ ,  we  may  put  a^ . 


104  ALGEBRA. 

For,  in  the  latter  of  these  expressions,  a  is  supposed  to  be 
resolved  into  twice  as  many  factors  as  in  the  former ;  and  the 
numerator  shows  that  twice  as  many  of  these  factors  are  to 
be  multiplied  together.  So  that  the  whole  value  is  not  al- 
tered, 

4      -      f 
Thus  a?^=j?®=a;^,  &:c.  that  is,  the  square  of  the  cube 

root  is  the  same,  as  the  fourth  power  of  the  sixth  root,  the 

sixth  power  of  the  9th  root,  &;c. 

J.  6  2n 

So  fl*=a^=a'^=a«.  For  the  value  of  each  of  these 
indices  is  2.     (Art.  135.) 

255.  From  the  preceding  article,  it  will  be  easily  seen,  that 
^  fractional  index  may  be  expressed  in  decimals . 

1.  Thus  a^  =a* "  or  a^*^  ;  that  is,  the  square  root  is  equal 
to  the  5th  power  of  the  tenth  root. 

i        JL?_ 

2.  a*  =a^ **  °  or  a**** *  ;  that  is,  the  fourth  root  is  equal  to 

the  25th  power  of  the  100th  root. 

3.  a^-a"""  5.  J  =  a^'^ 

4.  a^=ft=^**,  6.  a  ^  =a»-^^ 

In  many  cases  however,  the  decimal  can  be  only  an  ap- 
proximation to  the  true  index. 

X  1 

Thus  a^=a''*^  nearly.  aJ=a^  '^^'^^  very  nearly. 

In  this  manner,  the  approximation  may  be  carried  to  any 
degree  of  exactness  which  is  required. 

Thu^  a^=a^ -^^^^  a^'^=:a^ '"'''. 

These  decimal  indices  fonn  a  very  important  class  of  num- 
bers, called  logarithm^' 

It  is  frequently  convenient  to  vary  the  notation  of  powers 
pf  roots,  by  making  use  of  a  vinculum,  or  the  radical  sign  -/. 
In  doing  this,  wc  must  keep  in  mind,  that  the  power  of  a 
root  is  the  same,  as  the  root  of  a  power ;  (Art.  251,)  and  al- 
so, that  the  denominator  of  a  fractional  exponent  expresses 
a  root,  and  the  numerator,  a  power,     (Art.  250.) 

2  1  J 

Instead,  therefore,  of  a*,  we  may  write  («^)*,  or  {a^Y,  or 


EVOtUTlON.  iQ^ 

The  first  of  these  three  forms,  denotes  the  square  of  l}i& 
cube  root  of  a  ;  and  each  of  the  two  last,  the  cube  rooi  of 
the  square  of  a. 

And(6a;)^   =(6V)^=:=V^^. 


And  a 4-3/^-=  a +y    I   :=:^ a-i-y    . 


EVOLUTION. 

257.  Evolution  is  the  opposite  of  involution.  One  is 
finding  a  power  of  a  quantity,  by  multiplying  it  into  itself, 
The  other  is  finding  a  root,  by  resolving  a  quantity  into 
equal  factors.  A  quantity  is  resolved  into  any  number  of 
equal  factors,  by  dividing  its  index  into  as  many  equal  parts, 
(Art.  245.) 

Evolution  may  be  performed,  then,  by  the  following  gen? 
eral  rale  ^ 

Divide  the  index  ojf  the  qitantiti^,  by  the  number  ex* 
pressing  the  root  to  be  found. 

Or,  place  over  the  quantity  the  radical  sign  belonging  to 
the  required  root. 

1.  Thus  the  cube  root  of  a^  is  a-.  For  as  xa^  xa^=a^* 
Here  6,  the  index  of  the  given  quantity,  is  divided  by  3,. 

the  number  expressing  the  cube  root. 

J, 

2.  The  cube  root  of  a  or  a ^,  is  a ^  or  ^a, 

X         1          L 

For  a'^xa'''  xa%  or  l/axl^ax^a=a,  (Arts. 24^, 246  ) 

1  

3.  The  5th  root  of  ab,  is  (ab)  *  or  ^ah. 

4.  The  nth  root  of  a^,  is  «»  or  ^a^. 

5.  The  7th  root  of  2J-a:,  is  {^d-x^  or  l/^d^'xl 

6.  The  5th  root  of  a— a?| ,  isa— a:|*  or'^  a—7\  . 

1  1 

r.  The  cube  root  of  a^,  is  a^.     (Art.  163.) 

8.  The  4th  root  of  «-i  is  ar^ . 

9.  The  cube  root  of  a  ^  is  a  * . 

rn. 

10.  The  nth  root  of  a;^".  h  x", 

15 


106  ALGEBRA. 

258.  According  to  the  rule  just  given,  the  cube  root  of 
the  square  root  is  found,  by  dividing  the  index  |  by  3,  as  in 
example  7th.  But  instead  of  dividing  by  3,  we  may  multi- 
pli/hyi.     For  J~3=i~-J  =  -|x4.     (Art.  162.) 

Ill 

So  — ~w= — X — .     Therefore  the  mih  root  of  the  nth 
m  m       n 

-X- 
root  of  a  is  equal  to  a  "     "*. 


JLl  i  y  I-         i 

That  is,  a  "J    =0"     '»'=&""'. 

Here  the  two  fractional  indices  are  reduced  to  one  by 
multiplication. 

It  is  sometimes  necessary  to  reverse  this  process  ;  to  resolve 
an  index  into  two  factors, 
1 

Thusa;^=A'^^^=:a:'^l       That  is,  the  8th  root  of  x  itr 
equal  to  the  square  root  of  the  4th  root. 


So    a+Ar=«4-6r^"=a  +  6r 

It  may  be  necessary  to  observe,  that  resolving  the  index 
into  factors,  is  not  the  same  as  resolving  the  quantity  into 
factors.  The  latter  is  effected,  by  dividing  the  index  into 
parts. 

259.  The  rule  in  art.  257,  may  be  applied  to  every  case 
in  evolution.  But  when  the  quantity  whose  root  is  to  be 
found,  is  composed  of  several  factors^  there  will  frequently 
be  an  advantage  in  taking  the  root  of  each  of  the  factors 
separately. 

This  is  done  upon  the  principle,  that  the  root  of  the  pro- 
duct of  several  factors,  is  equal  to  the  product  of  their  roots. 

Thus  y/ab=^ax^h.     For  each  member  of  the  equation, 
if  involved,  will  give  the  same  power. 
The  square  of  -^/aO  is  ah,     (Art.  241.) 
The  square  of  y/a  X  yjb,  is  ^aX\/axVbx  yfh,    (Art.  102.) 
ButV«XV«  =  «-     (Art.  241.)     AniS.y/bX'\/h=h, 
Therefore  the  square  of  y/a  X  y/h=y/ax^ax  ^/hX^/h=ab^ 

which  is  also  the  square  of  y/ah, 

i      1  i 
On  the  same  principle,  (a^)"=a"6". 


'i  EVOLUTION.  107 

When,  therefore,  a  quantity  consists  of  several  factors,  we 
may  either  extract  the  root  of  the  whole  together;  or  we 
may  find  the  root  of  the  factors  separately,  and  then  multiply 
them  into  each  other. 

Ex.  1.  The  cube  root  of  xy^  is  either  [xyY  or  x^  y^. 

2.  The  5th  root  of  ^y,  is  \/Zy  ov  ^3x  V^' 

1  4-  J-  i 

3.  The  6th  root  of  abh,  is  {ahKf ,  ora^b^h^. 

1  ' 

4.  The  cube  root  of  8b,  is  (Sby  ^  or  2&"^. 

5.  The  nth  root  of  *x''y,  is,  '(x'^yY  or  ^". 

260.  The  root  of  a  fraction  is  equal  to  the  root  of 
the  numerator  divided  by  the  root  of  the  denominator. 

JL  12. 

a      a*  a^     a^      a 

1.  Thus  the  square  root  of -T- =1     For  — x-r— r". 

6      6^  b'^     bi     ^ 

i  i         X 

a      a"  a"*     a''  a 

2.  So  thenfhroot  of  T-=~T«  For -j  X  "T  *••  n  times  = -r-. 

mi  ,    .^^    .      V^  ^      /«^      A/ii^ 

3.  The  square  root  of—,  is  —=^.  4.  ^/__.-Ji--. 

«3/        Vfl^  "^x^y      Vxy 

261.  For  determining  what  sign  to  prefix  to  a  root,  it  is 
important  to  observe,  that 

An  odd  root  of  any  quantity  has  the  same  sign  as  THE 
QUANTITY  ITSELF  ,' 

An  EVEN  ROOT  OF  AN  AFFIRMATIVE  QUANTITY  IS  AMBIG- 
UOUS 5 

An   EVEN  ROOT  OF  A  NEGATIVE  QUANTITY  IS  IMPOSSIBLE. 

That  the  3d,  5th,  7th,  or  any  other  odd  root  of  a  quantity 
must  have  the  same  sign  as  the  quantity  itself,  is  evident 
from  art.  219. 

262.  But  an  even  root  of  an  affirmative  quantity,  may  be 
either  aftirmative  or  negative.  For  the  quantity  may  be 
produced  from  the  one,  as  well  as  from  the  otfaer.  (ATt. 
219.) 

Thus  the  square  root  of  a*  is  -\-a  or  —-a. 


iOS  ALGEBRA. 

An  even  root  of  an  affirmative  quantity  is,  therefore,  siaid' 
to  be  ambiguous,  and  is  marked  with  both  +  and  — . 

Thus  the  square  root  of  36,  is  ]^  V  36^ 
The  4th   root  of  x,  h'^x^. 

The  ambiguity  does  not  exi^t,  however,  when,  from  the 
nature  of  the  case,  or  a  previous  multipHcation,  it  is  known 
whether  the  power  has  actually  been  produced  from  a  posi- 
tive, or  from  a  negative  quantity.     Sec  art.  299. 

263.  But  no  even  root  of  a  negative  quantity  can  be  found. 
The  square  root  of  —a^  is  neither  +«  nor  —a. 
For  4-«X +«  =  +  «*.         And —ax —«=+«*  also. 

An  even  root  of  a  negative  quantity  is,  therefore,  said  to 
be  impossible  or  imaginary. 

There  are  purposes  to  be  answered,  however,  by  applying 
the  radical  sign  to  negative  quantities.  The  expression  y/  —  a 
is  often  to  be  found  in  algebraic  processes.  For,  although 
we  are  unable  to  assign  it  a  rank,  among  either  positive  or 
negative  quantities  ;  yet  we  know  that  when  multiplied  in- 
to itself,  its  product  is— «,  because  ^ — a  is  by  notation  a  root 
of  —a,  that  is,  a  quantity  which  multiphed  into  itself  pro- 
duces —  a. 

This  may,  at  first  view,  seem  to  be  an  exception  to  the 
general  rule  that  the  product  of  two  negatives  is  afiirmative. 
But  it  is  to  be  considered,  that  -^  -^a  is  not  itself  a  nega- 
tive quantity,  but  the  root  of  a  negative  quantity. 

The  mark  of  subtraction  here,  must  not  be  confounded 
with  that  which  \%  prefixed  to  the  radical  sign.  The  expres- 
sion ^J —a  is  not  equivalent  to  —^/a.  The  former  is  a  root 
of  —a  ;  but  the  latter  is  a  root  of  -}-« : 

For  — v/«X  — v^«=-l-  -^aa^^d. 

The  root  of  —a,  however,  may  be  ambiguous.  It  may  be 
either  +"/—«?  or  —■/—«. 

One  of  the  uses  of  imaginary  expressions,  is  to  indicate 
an  impossible  or  absurd  supposition,  in  the  statement  o{  a 
problem.  Suppose  it  be  required  to  divide  the  number  1 4 
into  two  such  parts,  that  their  product  shall  be  60.  If  one 
of  the  parts  be  x,  the  other  will  be  14— jr.  And  by  tl*e 
supposition 

^X(14-:r)=60,  or  Ma;*-.^^  :;=i60. 


EVOLUTION*  109 

This  reduced,  by  the  rules  in  the  following  section,  wil! 
give  x=7+  v^  — 11. 

As  the  value  of  x  is  here  found  to  contain  an  imaginary 
expression,  we  infer  that  there  is  an  inconsistency  in  the 
gtatement  of  the  problem  :  that  the  number  14  cannot  be 
divided  into  any  two  parts  whose  product  shall  be  60.^ 

264.  The  methods  of  extracting  the  roots  of  compound 
quantities  are  to  be  considered  in  a  future  section.  But 
there  is  one  class  of  these,  the  squares  of  binomial  and  re- 
sidual quantities,  which  it  will  be  proper  to  attend  to  in  this 
place.  It  has  been  shown,  (Art.  214,)  that  the  square  of  a 
binomial  quantity  consists  of  three  terms^  two  of  which  are 
complete  powers,  and  the  other  is  a  double  product  of  the 
roots  of  these  powers.     The  square  of  a-{-h^  for  instance,  is 

two  terms  of  which,  «*  and  6^,  are  complete  powers,  and 
2«&  is  twice  the  product  of  a  into  6,  that  is,  of  the  root  of  «^ 
into  the  root  of  6^. 

Whenever,  therefore,  we  meet  with  a  quantity  af  this  de- 
scription, we  may  know  that  its  square  root  is  a  binomial  ; 
and  this  may  be  found,  by  taking  the  root  of  the  two  terms 
which  are  complete  powers,  and  connecting  them  by  the 
sign  + .  The  other  term  disappears  in  the  root.  Thus,  to 
find  the  square  root  of 

take  the  root  of  a?^,  and  the  root  of  ?/*,  and  connect  them 
by  the  sign  +.     The  binomial  root  will  then  be  x-\-y. 

In  a  residual  quantity,  the  double  product  has  the  sign  — 
prefixed,  instead  of  +.  The  square  of  a—h,  for  instanc^^ 
is  a*— 2a6-f6^.  (Art.  214.)  And  to  obtain  the  root  of  a 
quantity  of  this  description,  we  have  only  to  take  the  roots 
of  the  two  complete  powers,  and  connect  them  by  the  sign 
— .     Thus  the  square  root  of  ^^  —  2xy-\-y^  is  x—y.     Hence, 

^Q5,  To  EXTRACT  A  BINOMIAL  OR  RESIDUAL  SQUARE  ROOT. 
TAKE  THE  ROOTS  OF  THE  TWO  TERMS  WHICH  ARE  COMPLETE 
POWERS,  AND  CONNECT  THEM  BY  THE  SIGN,  WHICH  IS  PREFIX- 
ED TO  THE  OTHER  TERM. 

Ex.   1.  Find  the  root  of  a:2+^ 4-1. 

The  two  terms  which  are  complete  powers  are  x^  and  1 . 
Their  roots  are  x  and  1.     (Art.  248.) 
The  binomial  root  is,  there forc>  x-\-\. 

*  &ee  Note  F. 


no  ALGEBRA. 

2.  The  square  root  of   a:^— 2a:-f  1,  is  a*  —  l.     (Art.  214.) 

3.  The  square  root  of  flS+G+i,  is  a-\~-l.     (Art.  224.) 

4.  The  square  root  of  a* +^0 -4- 1,  is  a -f  I . 

5.  The  square  root  of  a^ +«&+—,  is  «+'^. 

6.  The  square  root  of  «^4-"r  +  "^j  is  a+ — . 

266.  A  ROOT  WHOSE  VALUE  CANNOT  BE  EXACTLY  EXPRES- 
SED IN  NUMBERS,  IS  CALLED  A  SURD. 

Thus  yf2,  is  a  surd,  because  the  square  root  of  2  cannot 
be  expressed  in  numbers,  with  perfect  exactness. 
In  decimals,  it  is  1.41421 350  nearly. 

But  though  we  are  unable  to  assign  the  value  of  such  a 
quantity  when  taken  alone,  yet  by  multiplying  it  into  itself,  or 
by  combining  it  with  other  quantities,  we  may  produce  ex- 
pressions whose  value  can  be  determined.  There  is  there- 
fore a  system  of  rules  generally  appropriated  to  surds.  But 
as  all  quantities  whatever,  when  under  the  same  radical  sign, 
or  having  the  same  index,  may  be  treated  in  nearly  the  same 
manner  ;  it  will  be  most  convenient  to  consider  them  togeth- 
er, under  the  general  name  of  Radical  Quantities  ;  under- 
standing by  this  term,  every  quantity  which  is  found  under  a 
radical  sign,  or  which  has  a  fractional  index. 

267.  Every  quantity  which  is  not  a  surd,  is  said  to  be  ra- 
tional. But  for  the  purpose  of  distinguishing  between  radi- 
cals and  other  quantities,  the  term  rational  will  be  applied, 
in  this  section,  to  those  only  which  do  not  appear  under  a 
radical  sign,  and  which  have  not  a  fractional  index. 

REDUCTION  OF  RADICAL  QUANTITIES. 

268.  Before  entering  on  the  consideration  of  the  rules  for 
the  addiiion,  subtraction,  multiplication,  and  division  of  rad- 
ical quantities,  it  will  be  necessary  to  attend  to  the  methods 
of  reducing  them  from  one  form  to  another. 

First,  io  reduce  a  rational  quantity  to  the  form  of  a  rad- 
ical ; 

Raise  THE  quantity  to  a  power  of  the  same  name  as 

THE  GIVEN  ROOT,  AND  THEN  APPLY  THE  CORRESPONDING  RADI- 
CAL SIGN  OR  INDEX. 


RADICAL  QUANTITIES.  {[^ 

»Ex.  1.  Reduce  a  to  the  form  of  the  nth  root. 
The  nth  power  of  a  is  a\     (Art.  211.) 
Over  this  place  the  radical  sign,  and  it  becomes  ^  a". 
It  is  thus  reduced  to  the  form  of  a  radical  quantity,  with^ 

eut  any   alteration  of  its  value.     For  ^a^=:a'^  =rt. 

2.  Reduce  4  to  the  form  of  the  cube  root. 

Ans.  X/64:  or  (64)^. 

3.  Reduce  3a  to  the  form  of  the  4th  root. 

Ans.  ^81«^. 

4.  Reduce  ^ab  to  the  form  of  the  square  root. 

Ans.  i^a^b^)^. 


5.  Reduce  3  X  a— a;  to  the  form  of  the  cube  root. 

^  Ans.  ^27  x'a^l ^ .     See  art.  2 1 2. 

6.  Reduce  «2  to  the  form  of  the  cube  root. 
The  cube  of  a^  k  a^ .     (Art.  220.) 

And  the  cube  root  of  a^  is  l/a^  =a®  |*. 

In  cases  of  this  kind,  where  a  power  is  to  be  reduced  to 
the  form  of  the  nth  root,  it  must  be  raised  to  the  nth  power^ 
not  of  the  given  letter,  but  of  the  power  of  the  letter. 

Thus  in  the  example,  a^  is  the  cube,  not  of  a,  but  of  a*. 

7.  Reduce  a ^6^  to  the  form  of  the  square  root. 

8.  Reduce  d^  to  the  form  of  the  nth  root. 

269.  Secondly,  to  reduce  quantities  which  have  different 
indices,  to  others  of  the  same  value  having  a  common  index ; 

1 .  Reduce  the  indices  to  a  common  denominator ; 

2.  Involve  each  quantify,  to  the  power  expressed  by  the 
numerator  of  its  reduced  index. 

3.  Take  the  root  denoted  by  the  conmion  denominator. 

Ex.  1.  Reduce  a*  and  h^'  to  a  common  index. 

1st.  The  indices  ^  and  ~  reduced  to  a  common  denomin- 
ator, are  -^^  and  j\,     (Art.  146.) 

2d.  The  quantities  a  and  b  involved  to  the  powers  expres 
sed  by  the  two  numerators,  are  a^  and  6-. 


112  ALGEBllA, 

3(1.  The  root  denoted  by  the  common  denominator  is  -/^ 
The  answer,  then,  'is   a^ \'^^ 2indb^\'^'^ . 

The  two  quantities  are  thus  reduced  to  a  common  indeiL. 
•witliout  any  alterationJin  their  values. 


-3_ 


For  by  art.  254,  «"==«'%  which  by  art.  258,  ^a'^U^ , 

And  universally  a"  =«"»"  =  a^\  »?«. 

1  I 

2.  Reduce  a  ^  and  bx    to  a  common  index. 

The  indices  reduced  to  a  common  denominator  are   | 


and  f 


The  quantities^  then,  are  a^  and  (bxY,  or  a^|^  and  6*a;*|«. 

JL  ±  ± 

3*  Reduce  a^  and  6 "•     Ans.  a^'T  and  ^". 


4.%.Reduce  a?"  and  y .     Ans.  a?'"  1*""  and  ^" 
5.  Reduce  2=^  and  3^     Ans.  8*  and9\ 


i 

\mn 


TT© 


X -X 

3 2l» 


Q,  Reduce  («+&)^  and  (a?— ^)^.  Ans.  a +  6  |  anda?— y 

XX  2  X 

7.  Reduce  a^  and  b  ^.  8.  Reduce  cc^  and  5^. 

270.  When  it  is  required  to  reduce  a  quantity  to  a  given 
jhdex ; 

Divide  the  index  of  the  quantity  by  the  given  index,  place 
the  quotient  over  the  quantity,  and  set  tlie  given  index  over 
the  whole. 

This  is  merely  resolving  the  original  index  into  t\yo  fac- 
tors, according  to  art  258. 

i_ 
Ex.  1.  Reduce  «*  to  the  index  j. 

Byart.  162,  J-^J=ixl=f=i. 
This  is  the  index  to  be  placed  over  a,  which  then  becomtiji 

rr' ;  and  the  given  index  set  over  this  makes  it  a"^  i  ,  the   an- 
swer: 

2.  Reduce  a^  and  a?^,  to  the  common  index  f. 
2-^^=2x3=6,  the  first  index 
|-r-^=|X3=|,  the  second  index 
X  £  X       , 

Therefore  (a^y  and  {x^y  are  the  quantities  required. 


RADICAL  QUANTITIES.  113 

3.  Reduce  4^  and  3'^,  to  the  common  index  ^-.  , 
Answer.  {4^y  and  {3^y, 

271.  Thirdly,  to  remove  a  part  of  a  root  from  under 
the  radical  sign  ; 

If  the  quantity  can  be  resolved  into  two  factors,  one  of 
which  is  an  exact  power  of  the  same  name  with  the  root ; 

riND    THE  ROOT  OF  THIS  POWER,  AND  PREFIX  IT  TO  THE  OTH- 
ER FACTOR,  WITH  THE  RADICAL  SIGN  BETWEEN  THEM. 

This  rule  is  founded  on  the  principle,  that  the  root  of  the 
product  of  two  factors  is  equal  to  the  product  of  their  roots. 
(Art.  259.) 

It  will  generally  be  best  io  resolve  the  radical  quantity  in- 
to such  factors,  that  one  of  them  shall  be  the  greatest  power 
which  will  divide  the  quantity  without  a  remainder.  If 
there  is  no  exact  power  which  will  divide  the  quantity,  the 
reduction  cannot  be  made. 

Ex.  1.  Remove  a  factor  from  v'S. 
The  greatest  square  which  will  divide  8  is  4. 
We  may  then  resolve  8  into  the  factors  4  and  2.  For  4x2=8. 

The  root  of  this  product  is  equal  to  the  product  of  the 
roots  of  its  factors  ;  that  is,  •\/8=  V4X'v/2. 

But  '/4=2.  Instead  of  •v/4,  therefore,  we  may  substitute 
its  equal  2.     We  then  have  2x^/2ov  2y/2. 

This  is  commonly  called  reducing  a  radical  quantity  to  its 
most  simple  terms.  But  the  learner  may  not  perhaps  at  once 
perceive,  that  2/2  is  a  more  simple  expression  than  V^. 

2.  Reduce    ^/a^x.       Ans.  -y/a^  x  x^x^axV^^^^  V^' 

3,  Reduce    -^/Ts.        Ans.  ^9x2=  ^9x^2=3/2. 


4.  Reduce   ^64b^c.       Ans.  ^64^3x^^=46^^ 

4    la'^b  a  ^    lb 

5.  Reduce     V^T      Ans.—    y —j.     (Art.  260.) 


6.  Reduce  V«"^-     Ans.  a  V^,  or  ah\ 

7.  Reduce  (a3—a2&)^      Ans.  a  {a—by. 

8.  Reduce  (54a«6)^.        Ans.  3a^{2by. 

9.  Reduce    V98a^        10.  Reduce  Va^-^a^h' 

16 


114  ALGEBRA. 

272.  By  a  contrary  process,  the  co-efficient  of  a  radical 
quantity  may  be  introduced  under  the  radical  sign, 

1.  Thus«  V^="Va"^- 

Fora=  "-/a"  or  a".      Art.  253.)     And  ^a^x  V^=V^"^- 

Here  the  co-efficient  a  is  first  raised  to  a  power  of  the 
same  name  as  the  radical  part,  and  is  then  introdb  ced  as  a 
factor  under  the  radical  sign. 


2.  aix-hy  =:{a^  xx-^-b)^  =:.{a^x-a^hY, 


4. 


ADDITION  AND   SUBTRACTION  OP  RADICAL 
QUANTITIES. 

273.  Radical  quantities  may  be  added  like  rational  quan- 
tities, by  writing  them  one  after  another  with  their  signs.  (Art. 
69.) 

Thus  the  sum  of  -^a  and  ^b,  is  y/a-\-y/b. 

And  the  sum  of  ff2— /i?  and  a;'^— 2/",isa^  — A^-j-x^—y  ". 

But  in  many  cases,  several  terms  may  be  reduced  to  one? 
as  in  arts.  72  and  74. 

The  sum  of  2  -/«  and  2y/a  is  2  '^a-\-3'^a=5^a. 
For  it  is  evident  that  twice  the  root  of  «,  and  three  times 
the  root  of  a,  are  five  times  the  root  of  a,     Hence^ 

274.  When  the  quantities  to  be  added  have  the  same  rad- 
ical part,  under  the  same  radical  sign  or  index  ;  add  the  ra- 
tional parts,  and  to  the  sum  annex  the  radical  parts. 

If  no  rational  quantity  is  prefixed  to  the  radical  sign,  1  is. 
always  to  be  understood.     (Art.  244.) 


To     2V«y 
Add      l/ay 

5y/a 

-2v/a 

3{x  +  hy 

4{^+hy 

5bh^               aVb-h 
7bh^              yy/b-h 

Sum    3V«y 

7{x+hy 

(a-\-y)xVb-h 

^31518/?;^: 


RADICAL  QUANTITIES. . 

275.  If  the  radical  parts  are  originally  difFjp  "^  ^^ 
sometimes  be  made  alike,  by  the  reductions  i] 
articles. 

1.  Add  -/S  to  y/50.  Here  the  radical  parts  c.^a.  aot  the 
same.  But  by  the  reduction  in  art.  271,  -v/8=2'/2,  and 
^50=5v2.     The  sum  then  is  7^2. 

2.  AMy/Uhio  V46.         Ans.  V6  +  2v/5=6  V^. 

3.  Add^ft^^  to  y/h^x,    Al\^,a^/X'\-b^  ■y/x^{eL-\-h^)x  y/x. 

4.  Add  (SGa^yY  to  (25^)^     Ans.  {Ga-^5)  x/. 

5.  Add   VI  8a  to  3  V2o!. 

276.  But  if  the  radical  parts,  after  reduction,  are  different^ 
or  have  different  exponents,  they  cannot  be  united  in  the 
same  term ;  and  must  be  added  by  writing  them  one  after 
the  other. 

The  sum  of  Sy/l  and  2^^,  is  3v/6+2v/a* 

It  is  manifest  that  three  times  the  root  of  6,  and  twice  the 
root  of  «,  are  neither  five  times  the  root  of  &,  nor  five  times 
the  root  of  a,  unless  b  and  a  are  equal. 

The  sum  of  \^a  and  ^«,  is  %/a-\-\/a. 

The  square  root  of  «,  and  the  cube  root  of  «,  are  neither 
twice  the  square  root,  nor  twice  the  cube  root  of  «. 

277.  Subtraction  of  radical  quantities  is  to  be  performed 
in  the  same  manner  as  addition,  except  that  the  signs  in  the 
subtrahend  are  to  be  changed  according  to  art.  82. 


From     y/ay     4     ^a-\-x 
Sub.     S^ay     3     Va-f-x 

3F 
8A* 

a[x^y) 
b{x^y) 

-a     " 

-T-2«     " 

'D'\&.-2y/ay 

T 

a     " 

From  ^50,  subtract  VS.   Ans.5v'2-2  V2=3v/2.  (Art.275.) 
From  X/b'^y^  subtract  \/hy^,     Ans.  {b—y)x  y/hy. 
From  \/x,  subtract  ^x. 

MULTIPLICATION  OP  RADICAL  QUANTITIES. 
278.    Radical  quantities  may  be  multiplied,  like  other 


X 1$  ALGEBRA. 

quantities,  by  writing  the  factors  one  after  another,  either 
with  or  without  the  sign  of  multiplication  between  them, 
(Art.  93.) 

Thus  the  product  of  ^a  into  -y/^j  is  Va  X  y/h, 

1.  ill. 

The  product  of  h^  into  ^^  is  h^y^. 

But  it  is  often  expedient  to  bring  the  factors  under  the 
same  radical  sign.  This  may  be  done,  if  they  are  first  redu- 
ced to  a  common  index. 

Thus  V^X  \/2/=V^y»  For  the  root  of  the  product  of 
several  factors  is  equal  to  the  product  of  their  roots.  (Art. 
259.)     Hence, 

279.  Quantities  under  the  same  radical  sign  or  in- 
dex, MAY  BE  multiplied  TOGETHER  LIKE  RATIONAL  QUANTI- 
TIES, THE  PRODUCT  BEING  PLACED  UNDER  THE  COMMON  RADI- 
CAL SIGN  OR  INDEX.* 

J-  -L 

Multiply  ^x  into  ^?/,  that  is,  x^  into^^. 

The  quantities  reduced  to  the  same  index,  (Art.  269,)  are 

XI  i       6     . 

{x^Y,  and  (t/2)%  and  their  product  is  (x^fY  =    Va;'^*. 


Mult. 

Va  +  m 

Into 

Va  —  m 

Prod. 

Va^  -m^ 

(a^xY  (a"*"") 


I 

win 


Multiply    /8uc6  into   -v/2x6.     Prod.  v'16a;2  62=4a?6. 

In  this  manner  the  product  of  radical  quantities  often  be- 
ciomes  rational. 

Thus  the  product  of  V2  into  V18=V36=6. 

1  i  -1 

And  the  product  of  {a^y^Y  ^^^^  {(^^yY  =(«*y*)*=a^. 

280.  Roots  of  the  same  letter  or  quantity  may  be 
multiplied,  by  adding  their  fractional  exponents. 

The  exponents,  like  all  other  fractions,  must  be  reduced 
to  a  common  denominator,  before  they  can  be  united  in  one 
term.     (Art.  148.) 

*  The  case  of  an  imaginary  root  of  a  negative  quantity  may  be  consid- 
ered an  exception.     (Art.  263.) 


RADICAL  QUANTITIES.  11^' 

Thus  a^x«^=a^^^=a«^^=a\ 

The  values  of  the  roots  arc  not  altered,  by  reducing  their 
indices  to  a  common  denominator.     (Art.  254.) 

Therefore  the  first  factor  «~  =a^( 

1        2  ( 
And  the  second  a^  =a^3 

But  tt^=««  xJx a'.     (Art.  250.) 


4^4*4-4 


^-J 


The  product  therefore  is  a^  xa^  Xa^  Xa^  Xa^  =«^ . 

And  in  all  instances  of  this  nature,  the  common  denomiw- 
ator  of  the  indices  denotes  a  certain  root ;  and  the  sum  of 
the  numerators  shows  how  often  this  is  to  be  repeated  as  a 
factor  to  produce  the  required  product. 

i-         ±  m  2^         n»+n 

Thus  a"xa"*=a"»^Xa'"»=a«^. 

Mult.     32^^    ■     a^xJ       {a  +  hf       {ci-yY  a;""^ 

Into         y^         a^  {a-^hf       {a-yf  -""* 


X 


Prod.      3^'^  {a-\-hy  '  x     ''' 


The  product  of  ^^  intoy     ^  is  3/^        =y  . 
The  product  of  a"  into  a~  ",  is  a"  """=«<>  =  1. 
And  a^«""^x^^^''=/*"~''"^^~^=a:«=l. 

The  product  of  a^  into  a^=a^Xa^=a^. 

281.  From  the  last  example,  it  will  be  seen,  that  powers 
and  roots  may  be  multiplied  by  a  common  rule.  This  is  one 
of  the  many  advantages  derived  from  the  notation  by  frac- 
tional indices.  Any  quantities  whatever  may  be  reduced  to 
the  form  of  radicals,  (Art.  268,)  and  may  then  be  subjected 
to  the  same  modes  of  operation. 

Thus  ?/ 3  X  2^6  =  ?/ ^  "^  ^ = i^~«~. , 

i  4.JL         n+l 

And  a;Xa;"=.T*     "=a;  «. 


IIB 


ALGEBRA. 


Tlie  product  will  become  rational,  whenever  the  numera- 
tor of  the  index  can  be  exactly  divided  by  the  denominator. 


1  2  12 

Thus  «^xa^Xa^=a^  =«* 


And  {a+byx{a+b)     ^=(a'{-by  =a-\-b. 

3  2  5 

And  a^  x  a^  =a^  =«. 

282.  When  radical  quantities  which  are  reduced  to  the 
same  index,  have  rational  co-efficients,  the  rational 

PARTS  MAY  BE  MULTIPLIED  TOGETHER,  AND  THEIR  PRODUCT 
PREFIXED  TO  THE  PRODUCT  OF  THE  RADICAL  PARTS. 

1.  Multiply  a^b  into  Cy/d, 

The  product  of  the  rational  parts  is  ac. 
The  product  of  the  radical  parts  is  y/bd. 
And  the  whole  product  is  ac^bd. 
For  ay/b  is  a  X  -y/i.      f  Art.  244.)     And  c^d  is  c  X  \/d. 

By  art.  102,  « X  \fb  into  c x ^/d,  is  ox  \/bxcX-\/d -,  or  by 
changing  the  order  of  the  factors, 

axcxVbxVd=acX\fbd=:acy/bd. 

2.  Multiply  ax^  into  bd^ , 

When  the  radical  parts  are  reduced  to  a  common  index, 

the  factors  become  a{x^y  and  h{d^y , 

The  product  then  is  ab  {x^d^f. 

But  in  cases  of  this  nature,  we  may  save  the  trouble  of 
reducing  to  a  common  index,  by  multiplying  as  in  art.  278. 

J-  JL  A      .1 

Thus  ax^  into  bd"  is  ax^bd'^. 


Mult. 

a(b  +  xy^         ay/y^ 

ciy/x              ax     ^      X  \/Z 

Into 

y{b-xY          b^hy 

by/x              by'^'^    y  3/9 

Prod. 

ayib--^x^f 

aby/x^  ^abx                        3xy 

283.  If  the  rational  quantities,  instead  of  being  co-effi- 
cients to  the  radical  quantities,  are  connected  with  them  by 
the  signs  ^f-  and  — ,  each  term  in  the  multiplier  must  be 
multiplied  into  each  in  the  multiplicand,  as  in  art.  100. 


RADICAL  QUANTITIES.  119 


Multiply  a-\-y/h 
Into  c4-  y/d 


ac-\-cy/h 

a^/d-\-yfhd 

aC'{^c^/b■\-ay/d'\-^Jhd. 
The  product  of  «+v'^  into  l-\-ry/if  is 

1.  Multiply  v/«into  ^^b.  Ans.  Va^P, 

2.  Multiply  V5  into  3-v/8.  Ans.  30v/T0- 

3.  Multiply  2v/3  into  3  ^4.  Aiis.  6\/432. 


4.  Multiply  ^d  into    V«6.  Ans.    Va^^M^. 

5.  Multiply  J^±uitoJ^  Ans.  J^- 

6.  Multiply  a(a— J?) *  into  (c— ^)x  (a;^)^. 

Ans.  {ac—ad)x{a^x—ax^)^. 

DIVISION  OF  RADICAL  QUANTITIES. 

284.  The  division  of  radical  quantities  may  be  expressed, 
by  writing  the  divisor  under  the  dividend,  in  the  form  of  a 
fraction. 

Thus  the  quotient  of  ^a  divided  by  V^,  is  -rr-. 

1 
And  («+/if  divided  by  (6+:^)"  is^^±^. 

In  these  instances,  the  radical  sign  or  index  is  sepaMely 
applied  to  the  numerator  and  the  denominator.  But  if  the 
divisor  and  dividend  are  reduced  to  the  same  index  or  radi- 
cal sign,  this  may  be  applied  to  the  whole  quotient. 

_  Z^a     ^   fa 

Thus  l^a^l^b=~T=  yy.  For  the  root  of  a  frac- 
tion is  equal  to  the  root  of  the  numerator  divided  by  the 
root  of  the  denominator.     (Art.  260.) 


120  ALGEBRA. 

Again,  ya6-7-!^6  =  !J/a.    For  the  product  of  this  quotient 
into  the  divisor  is  equal  to  the  dividend,  that  is, 
Va  X  :y i  =  **  ^Jah,     Hence, 

285.  Quantities  under  the  same  radical  sign  or  in- 
dex, MAY  BE  DIVIDED  LIKE  RATIONAL  QUANTITIES,  THE  QUO- 
TIENT BEING  PLACED  UNDER  THE  COMMON  RADICAL  SIGN  OR 
INDEX. 

Divide  (x^y^Y  \i^  «/  . 


These  reduced  to  the  same  index  are  (pc^^y^Y  and  (y^) 

1  3  X 

And  the  quotient  is  (a;^)^=a?^=a?^. 


Divide    Vea^o?  yl dhx^  {a'^^axy  {a^hY    .{a^y^Y 

By           ^/3x  Vdx  a"^  {axf*      {ayY 

i  1 

Quot.      V2a=^  {a^'^-xY  {ayY 


286.  A  ROOT  IS  DIVIDED  BY  ANOTHER  ROOT  OF  THE  SAME 
LETTER  OR  QUANTITY,  BY  SUBTRACTING  THE  INDE]^  OP  THE. 
DIVISOR  FROM  THAT  OF  THE   DIVIDEND. 

X  1  X— 1  ^ —It         t  i 

Thusa'-7-a^=a'     ^=a^     ^  =a^  =za\ 

X  3  1  1  1.  X 

For    a2=a^=a^Xa^  xa^  and  this  divided  by  a®  \$ 

i        ^        w  X         1         2.        i 

JL 

a' 

X  X         X X 

Jn  the  same  manner,  it  may  be  shown  that  a"*--  a\*::^a^    ". 

XX  2  m-^n  _2  I 

Divide  (3a)' ^       {axY       a«'»        (6+y)«        (^^^'^ 
By  (^  {axY       a^'        (b-^yY         (r^y^Y 


Quot.       {SaY  o"  (r^^')     ^ 


Powers  and  roo^5  may  be  brought  promiscuously  together,, 
and  divided  according  to  the  same  rule.     See  art,  281. 


RADICAL  QUANTITIES.  121 

Thus  a^ -ra"^=tt 2  ~^=:a^.     For  a^xa^=a^=a^ 

287.  When  radical  quantities  which  are  reduced  to  the 
same  index  have  rational  co-efpicients,  the  rational 

PARTS  MAY    BE  DIVIDED  SEPARATELY,  AND  THEIR  QUOTIENT 
PREFIXED  TO  THE  QUOTIENT  OF  THE  RADICAL  PARTS, 

Thus  ac-^/bd-^a  y/b=cy/d.     For  this  quotient  multiplied 
into  the  divisor  is  equal  to  the  dividend. 

I 


Divide   l^xyfay 

Udhyfhx 

hy{a^x^Y 

16v'32 

hy/xy 

By             6  ^Ja 

2hy/X 

y{axY 

8^/4 

y/y 

Axy/y  h{a^xY  b  y/x 


Divide  ab{xH)'^  hy  a(x)*. 
These  reduced  to  the  same  index  are  ab{x^bY  and  a{x^Y . 

X  X 

The  quotient  then  is  b{bY  =(6^^.     (Art.  272.) 

To  save  the  trouble  of  reducing  to  a  common  index,  the 
division  may  be  expressed  in  the  form  of  a  fraction. 

ab{x^bY 


T 

he  quotient  will  then  be           j.  . 

a{x-) 

1. 

Divide  2  l/bc  by  3  ^/ac. 

Ans.l     7a3e. 

2. 
3. 
4. 

Divide  \Ol/\0^hy  5   ^4. 
Divide  10V27by  2  V3. 
Divide  8v/108  by  2v/6. 

Ans.  2  3^^27  =  6 
Ans.  15. 
Ans.  12^2. 

5. 

DWnie{aH^d^Y  bj  ^^ 

Ans.  {abY* 

6.  Divide  (16«3_] 2^3^)3  ^,2«.  Ans.  (4a- 3a?) ^ 

INVOLUTION  OF  RADICAL  QUANTITIES. 

288.  Radical  quantities,  like  powers,  are  involved 
by  multiplying  the  index  of  the  root  into  the  index  of 
the  required  power. 
17 


j  22  ALGEBRA. 

1.  The  square  of  a ^=a^        =a*.     For  «=*  xa^=a^. 

J       ^X3       1  i       i       i       3 

2.  The  cube  of  a*  =a*        =a4.     For  ct^  xa*  xa*=a^. 

3.  And  universally,  the  nth  power  of  a^±=.d!'''       =.a^. 

i       i       i 
For  the  nth  power  of  d"'==a"'xa"' ,  , ,  ,  n  times,  and  the 

tt 
sum  of  the  indices  will  then  be  ^ . 

X     1  s    s 

4.  The  5th  power  of  a^  y'^,  is  a^y^.   Or,  by  reducing  the 
roots  to  a  common  index, 

i     i  3^     _3  3 

5.  The  cube  of  «  V"  is  a^x"'  or  (a"'x")^i. 

6.  The  square  of  aJx'^,  is  tf^A?*. 

1        -X3        ^ 
The  cube  of  a^  is  a^        =0^  =za. 

1  n 

And  the  nth  power  of  a",  is  «  «=a.     That  is, 

289.    A  ROOT  IS  RAISED    TO    A   POWER  OF    THE    SAME    NAME. 
BY  REMOVING  THE  INDEX  OR  RADICAL  SIGN. 


Thus  the  cube  of   Vh-^-x,  is  h-\-x. 

J. 
And  the  nth  power  of  («— ^)",  is  a—y. 

290.  When  the  radical  quantities  have  rational  co-efficients^ 
these  must  be  also  involved. 

1.  The  square  a^^x,  is  a^'i/x^. 
For  av'^XaV«=a'Va;^. 

2.  The  nth  power  of  a"V%  is  a«"»^'". 


3*  The  square  of  a  ^Jx—y,  is  a^  X  {x—y)» 
5.  The  cube  of  Sa^/y,  is  27a^y. 

291.  But  if  the  radical  quantities  are  connected  with  oth- 
ers by  the  signs  +  and  — ,  they  must  be  involved  by  a  mul- 
tiplication of  the  several  terms,  as  in  art.  213. 


EADICAL  QUANTITIES.  12; 

Ex.   1.  Required  the  squares  of  a+  y/y  and  a^-yjij. 
a-{-y/y  o.—yfy 

a-\-yJy  (^-■\fy 

a^  ■\-a  y/y  a^  —^y/y 

ay/y\-y  ^a^y^-y 


2.  Required  the  cube  of  a  —  y/h» 

3.  Required  the  cube  of  2c?+Va?» 


292.  It  is  unnecessary  to  give  a  separate  rule  for  the  evo- 
lution of  radical' quantities,  that  is,  for  finding  the  root  of  a 
quantity  which  is  already  a  root.  The  operation  is  the  same 
as  in  other  cases  of  evolution.  The  fractional  index  of  the 
radical  quantity  is  to  be  divided,  by  the  number  expressing 
the  root  to  be  found.  Or,  the  radical  sign  belonging  to  the 
required  root,  may  be  placed  over  the  given  quantity.  (Art. 
257.)  If  there  are  rational  co-efficients,  the  roots  of  these 
must  also  be  extracted. 

-   .      -~2       - 
Thus,  the  square  root  of  a^,  is  a^  '  '"=«^. 

1         1        i 
The  cube  root  of  a{xyy,  is  a^(xy)^. 

The  nth  root  of  a\/by,  is  "V  a  Vby, 

293.  It  may  be  proper  to  observe,  that  dividing  the  frac- 
tional index  of  a  root  is  the  same  in  effect,  as  multiplying  the 
number  which  is  placed  over  the  radical  sign.  For  this 
number  corresponds  with  the  denominator  of  the  fractional 
index ;  and  a  fraction  is  divided,  by  7nultiplying  its  denom- 
inator. 

Thus  ^a=a*.  \/a=a^ 

On  the  other  hand,  multiplying  the  fractional  index  is  e- 
quivalent  to  dividing  the  number  which  is  placed  over  the 
radical  sign. 

Thus  the  square  of  ^a  or  a®,  is  ^a  or  a^       =ai. 


124  ALGEBRA. 

293.6.  In  algebraic  calculations,  we  have  sometimes  oc- 
casion to  seek  for  a  factor,  which  multiplied  into  a  given 
radical  quantity,  will  render  the  product  rational.  In  the 
case  of  a  simple  radical,  such  a  factor  is  easily  found.  For 
if  the  nth  root  of  any  quantity,  be  multiplied  by  the  same 
root  raised  to  a  power  whose  index  is  ?i--],  the  product  will 
be  the  given  quantity. 

i  n--l  n 

Thus  :j/a:X  yx"~^  or  a:"  X  a:  «  =a:«  =a?. 

And  (a;+y)"x(a;-i-y)~«  =a;-f  y. 

So  y/aXy/a=a,     And  ^ax\/a^  =':\/a^=:^a. 

And  V«X  V«'=«j  <$^c.       And  fa-f  &)'' x(«-t-6)^=«4-6. 

And  (x+t/r  X(i:+2/J^=a:+y. 

293.C.  A  factor  which  will  produce  a  rational  product, 
when  multiplied  into  a  binomial  mrd  containing  only  the 
square  root,  may  be  found,  by  applying  the  principle,  that 
the  product  of  the  sum  and  difference  of  two  quantities,  is 
equal  to  the  difference  of  their  squares.  (Art.  235.)  The 
binomial  itself,  after  the  sign  which  connects  the  terms  is 
changed  from  -f  to  — ,  or  from  —  to  +?  will  be  the  factor 
required. 

Thus  (v'a+  y/h)xya-^/h)==  y/a^ -y/b'' -a-h,  which 
is  free  from  radicals. 

So  (1+V2)X(1— /2)  =  l-2=:-l. 
And  (3-2V2)x(3+2v/2)=il. 

When  the  compound  surd  consists  of  more  than  two  terms, 
it  may  be  reduced,  by  successive  multiplications,  first  to  a 
binomial  surd,  and  then  to  a  rational  quantity. 

'    Thus  (  VIO-  V2-^/3)x(  VIO-I-  V2+  yf3)^b-2y/Q,  a 
binomial  surd. 

And  (5~2V6)x(5  +  2/6)=l. 

Therefore  (x/ 10 -V2-\/3)  multiplied  into  (v/104-v'2\/-f3) 
X(5  +  2v/6)=l. 

293.C?.  It  is  sometimes  desirable  to  clear  from  radical  signs 
the  numerator  or  denominator  of  a  fraction.  This  may  be 
effected,  without  altering  the  value  of  the  fraction,  if  the 


RADICAL  QUANTITIES.  125 

iiumerator  and  denominator  be  both  multiplied  by  a  factor 
which  will  render  either  of  them  rational,  as  the  case  may 
require. 

1.  If  both  parts  of  the  fraction  j^  be  multiplied  by  y/a. 

y/aXy/a        a  ,  .  ,     , 

it  will  become   .     ■    ,  =~ — ,  m  which  the  numerator  is  a 

rational  quantity. 

Or  if  both  parts  of  the  given  fraction  be  multipHed  by  y/x, 

V  fix 

it  will  become ,  in  which   the  ^e?iowma/or  is  rational. 

.    rru    <■     ,•  ^^  b^x{a+x)^     b^X{a+xf 

2.  The  traction  j-= rXF= — — Z * 

3.  The  fraction  — ^~  = i = }' 


a          ax  ^         a\/a;""-^ 
4.  The  fraction  ~I="T — ;ii:r= • 

X 

y/2  -v/SxCS+V^         2+3^/2 

b.  The  fraction  ^^^^^  =(i -^2X3+^/2)'^       7      ' 

3  (3^/5  +  ^/2) 

6.  The  fraction  ^^3i^=(^5_  ^^^^  V5+V2)  =  ^^+^^- 

6       6x5*        6       

7.  Thefraction  — =-7— J-  =  T"*/l25• 
8.  The  fraction 

8_ 8x(v/3-V2-l)(- V2)       __ 

y3+v/2+l-(V3+v/2+l)(V3-V2-l)(-v/2)-^"^^^"^^^^- 

9.  Reduce  —rz  to  a  fraction  having  a  rational  denomin- 

ator. 

a  —  yfh 

10.  Reduce  ~~^Jl  to  a  fraction  having  a  rational  denom- 
inator. 

293.6.  The  arithmetical  operation  of  finding  the  proxi- 
joiate  value  of  a  fractional  surd,  may  be  shortened,  by  ren- 


126  ALGEBRA. 

dering  either  the  Numerator  or  the  denominator  rational. 
The  root  of  a  fraction  is  equal  to  the  root  of  the  numera- 
tor divided  by  the  root  of  tlie  denominator.     (Art.  260.) 

Thus  — ^=^^.      But  this  may  be  reduced  to  «.^  ^  Va""* 

or Y .     (Art.  293.<?.) 

a       y/a  a  yl  ah 

Ihe  square  root  of  -y-  is  -y,,  or-T=-,  or  ~t~. 

When  the  fraction  is  thrown  into  this  form,  the  process  of 
extracting  the  root  arithmetically,  will  be  confined  either  to 
the  numerator,  or  to  the  denominator. 

™       ,,  ■  .     3     V3     ^/3X^/7      VsT 

Thus  the  square  root  of  —  =:  — =  .^       .   =  — y— 

Examples  for  Practice, 

1.  Find  the  4th  root  of   81a^ 

2.  Find  the  6th  root  of  (a+b^K 

3.  Find  the  nth  root  of  {x  —rff* 

4.  Find  the  cube  root  of  — 126  a^x^. 

5.  Find  the  square  root  of  ^—^2* 

32a^x^^ 
G.  Find  the  5th  root  of       ^^^     . 

7.  Find  the  square  root  of  a;^  —Gbx  +  h^, 

8.  Find  the  square  root  of  a^  ■^ay-\--T. 

9.  Reduce  ax^  to  the  form  of  the  6th  root. 

10.  Reduce  —  3y  to  the  form  of  the  cube  root. 

n.  Reduce  a^  and  a^  to  a  common  index, 

12.  Reduce  4^  and  5^  to  a  common  index. 

I  i  -J 

13.  Reduce  a'^  and  5*  to  the  common  index  \ 

J.  1  1 

14.  Reduce  2^  and  4*  to  the  common  index  *. 


RADICAL  QUANTITIES.  22" 

15.  Remove  a  factor  from   V294. 


16.  Remove  a  factor  from    y/x^—a^x^  ,  ■     . 

17-  Find  the  sum  and  difference  of  Vl6a*a?andv'4a-a?, 

18.  Find  the  sum  and  difference  of  VTd2  and  V^i. 

19.  Multiply    iVlQ  into  5^/4. 

20.  Multiply  4+2^/2  into  2-  V2. 

21.  Multiply  a(a+y/cy  into  h{a'-"y/cf. 

22.  Multiply  2(a-f  6)"  into  3(a+^)'"- 

23.  Divide  6  ^54  hy  3  yf2. 

24.  Divide   4\/72  by  2^18. 

25.  Divide  y/1  by  ^7. 

26.  Divide   8^512  by  4^2. 

27.  Find  the  cube  of  17^21. 

28.  Find  the  square  of  5  +  ^/2. 

29.  Find  the  4th  power  of  1^/6. 

30.  Find  the  cube  of  y/x  —  y/h. 

31.  Find  a  factor  which  will  make  ^y/y  rational. 

32.  Find  a  factor  which  will  make  y/5  —  -^x  rational. 

33.  Reduce  —j-  to  a  fraction  having  a  rational  numerator. 

V6 

34.  Reduce     .^        ^  to  a  fraction  having  a  rational  de- 
nominator. 


.f^^- 


e-^^Ljt:^ 


SECTION  X. 


REDUCTION  OF  EQUATIONS  BY  INVOLUTION 
AND  EVOLUTION. 

Art.  294.  xN  an  equation,  the  letter  which  expresses  the 
unknown  quantity  is  sometimes  found  under  a  radical  sign. 
We  may  have  y/x—a. 

To  clfear  this  of  the  radical  sign,  let  each  member  of  the 
equation  be  squared,  that  is,  multiplied  into  itself.  We  shall 
then  have 

Va?X\/a?=a«,         Or,  (Art.  289,)  x-=a^. 

The  equality  of  the  sides  is  not  affected  by  this  operation, 
because  each  is  only  multiplied  into  itself,  that  is,  equal  quan- 
tities are  multiplied  into  equal  quantities- 

The  same  principle  is  applicable  to  any  root  whatever. 
If  ]j/a?=«  ;  then  «=a".  For  by  art.  289.  a  root  is  raised  to 
a  power  of  the  same  name,  by  removing  the  index  or  radi- 
cal sign.     Hence, 

295.  When  the  unknown  quantity  is  under  a  radical 

SIGN,  the    equation    IS  REDUCED  BY  INVOLVING  BOTH   SIDES, 

to  a  power  of  the  same  name,  as  the  root  expressed  by  the 
radical  sign. 

It  will  generally  be  expedient  to  make  the  necessary  trans- 
positions, before  involving  the  quantities ;  so  that  all  those 
which  are  not  under  the  radical  sign,  may  stand  on  one  side 
of  the  equation. 

Ex.  1.  Reduce  the  equation  v'a?4-4=9 

Transposing  +4  Va?=9  — 4=5 

Involving  both  sides  a;  =5*  =25. 

2.  Reduce  the  equation  a+!J/a;— 6=<? 

By  transposition,  \/x=^d^h—a 

By  involution,.  vT=(c?4-6— a)^ 


EQUATJONS.  129 

3*  Reduce  the  equation  v'cc+i=4 

Involving  both  sides,  a?-hl=4^=64 

-    And  x=63. 

4,  Reduce  the  equation  4+3^^3;— 4=6+ i 

Clearing  of  fractions,  8  +  6  Va;— 4  =  13 

And  Vac  — 4=1 

Involving  both  sides,  a; — 4 = || 

And  cc=|f  +  4. 

3-^d 


5,  Reduce  the  equation  ^^*  + V^""v^^a~jLr7~ 

Multiplying  by  Va^  +  ^x,     a^+^x^S+d 
And  V'''c=3+J-a* 

Involving  both  sides,  a?  =  (3 + d—a^  Y . 

In  the  first  step  in  this  example,  multiplying  the  first  mem- 
ber into  y/a^  +  ^x,  that  is,  into  itself,  is  the  same  as  squar- 
ing it,  which  is  done  by  taking  away  its  radical  sign.  The 
other  member  being  a  fraction,  is  multiplied  into  a  quantity 
equal  to  its  denominator,  by  cancelhng  the  denominator. 
(Art.  159.)  There  remains  ,a  radical  sign  over  x,  which 
must  be  removed  by  involving  both  sides  of  the  equation. 

6.  Reduce  3  + V«--|=6. 

7.  Reduce  4  /~=8. 

8.  Reduce  (2a^+3)"^  +  4=7. 


9.  Reduce    Vl2+aJ=2+v/a\ 


10.  Reduce  Vx--a=^a?— |\/a. 


11.  Reduce  V5X  Va?+2  =  2+  Vsx. 
x-^ax       y/X 


12.  Reduce 

13.  Reduce 


Va;+2B     ^^  +  38 
18 


Ans. 

^=4fi. 

Ans. 

a? =20. 

Ans. 

a?=12. 

Ans. 

a?=!4. 

Ans. 

25a 

Ans. 

9 

Ads. 

1 

Ans. 

^.==4, 

1 30  ALGEBRA. 

2a 

14.  Reduce   •v/^4-  '^a+a?=  ', .       Ans,  x=^a, 

Va+x 

2a2 


15,  Reduce  x-\-  V a^ -j-x^  =  ,        — — .  Ans.  A;=a/i. 


16.  Reduce  x-\-a=^^a^ -^xy/b^-^-x'^ .        Ans.  ic= 


&2-4a2 


40 


, 4  2 

17.  Reduce  V  2+a;  +  V^=-7r==^.  Ans.  a:  =  —. 


^^2  + 


0; 


3 


18.  Reduce  v/x— 32  =  16— V^.  Ans.  a:=81. 

19.  Reduce  V4x  +  17=2v'a;+l.  Ans.  a;  =  16. 

Vei— 2     4V6^9 

20.  Reduce  -7^=. —  =-— -nr: — .  Ans.  a:  =6. 

V6ar+2     ^VQx-\-& 

REDUCTION  OF  EQUATIONS  BY  EVOLUTIOIJf. 

29G.  In  many  equations,  the  letter  which  expresses  the 
Unknown  quantity  is  involved  to  some  power.  Thus  in  the 
equation 

a;2=16 
we  have  the  value  of  the  square  of  a:,  but  not  of  x  itself. 
If  the  square  root  of  both  sides  be  extracted,  we  shall  have 

a;=4. 

The  equality  of  the  members  is  not  affected  by  this  re- 
duction. For  if  two  quantities  or  sets  of  quantities  are 
equal,  their  roots  are  also  equal. 

If  (ac+ «)"=&-{- A,  then  x  +  a=:%/h-\-h.     Hence, 

297.  When  the  expression  containing  the  unknown* 
(quantity  is  a  power,  the  equation  is  reduced  by  ex- 
TRACTING THE  ROOT  OF  BOTH  SIDES,  a  TODt  of  the  Same  name 
as  the  power. 

Ex.  K  Reduce  the  equation  64-a?2— 8=7 

By  transposition  a;'^  =7  +  8  —  6  =  9 

By  evolution  a?=±^9i=±3. 

The  signs  +  and  —  are  both  placed  before  v'^,  because 
an  even  r»ot  of  an  affirmative  quantity  is  ambiguous*  (Art. 
261.) 


EQUATIONS.  131 

2.  Reduce  the  equation  5a;a  — 30-ar»  +  34 
Transposing,  &:c.  aj^—jg 

By  evolution,  a:  =±4. 

x^      ^      x^ 

3.  Reduce  the  equation      ^+'y~^~""^ 

bdh  —  abd 
Clearing  of  fractions,  &:o,     x  ^  =, — ,  ,    , 

-\- (ldh-abd\\ 
By  evolution,  ^  =  _V — h-^d    ) 

4.  Reduce  the  equation  a4-df2c"=10— a?^ 

lO-a 
Transposing,  &c.  «"=  ^  ,  ^ 

/10-aU 
By  evolution,  ^~V^+T/  * 

298.  From  the  preceding  articles,  it  will  be  easy  to  see  m 
what  manner  an  equation  is  to  be  reduced,  when  the  ex- 
pression containing  the  unknown  quantity  is  a  power,  and  at 
the  same  time  under  a  radical  sign  ;  that  is,  when  it  is  a  root 
of  a  power.  Both  involution  and  evolution  will  be  necessa- 
ry in  this  case. 


Ex.  I.  Reduce  the  equation 

t/^»=4 

By  involution 

a;2=4'=64 

By  evolution 

a:=±v/64=:±8. 

2.  Reduce  the  equation 

y/x'^-a^h-d 

By  involution 

x^--a=:h^-2hd+d* 

And 

x'^^zh^-^hd+d^+a 

By  evolution 

x=:%/h^--2hd+d^+a. 

3.  Reduce  the  equation 

Multiplying  by  (x-a^)  (Art.  279.)  {x^-a^f^rza^h 
By  involution  x'^  —a^  =a^  -^1ah->rh'^ 

Trans,  and  uniting  terms        x^  =2a^  4-2«64-i^ 

By  evolution  *  =  (2^2  4-  2a6+ 6«)^. 


XS2  ALGEBRA. 


Problems. 


Prob.  1.  A  gentleman  being  asked  his  age,  replied  ;  "  If 
you  add  to  it  ten  years,  and  extract  the  square  root  of  the 
sum.  and  from  this  root  subtract  2,  the  remainder  will  be  6." 
"WTiat  was  his  age  ? 


By  the  conditions  of  the  problem      V^rH- 10—2=6 
By  transposition  J  Va?+ 10=6  +  2=8 

By  involution,  a?+ 10=;  82=64. 

And  <r=64-10=54. 

Proof  (Art.  194.)  a/54T  10-2=6. 

Prob.  2.  If  to  a  certain  number  22577  be  added,  ai^d  the 
square  root  of  the  sum  be  extracted,  and  from  this  163  be 
subtracted,  the  remainder  will  be  237.  What  is  the  num- 
ber? 

Let  a;  =the  number  sought.  6  =  163 

a=22577  c=237. 

By  the  conditions  proposed         '^x+a—b=:c 
By  transposition,  V'^+«=c+6 

By  involution,  a?+a=(c+6)^ 

And  cc  =  {c+hy—a 

Restoring  the  numbers,  (Art.  52.)  ^=(237+163)^  -22577 
That  is,  0?  =  160000-22577  =  1 37423. 

Proof  V 1 374234  22577-1 63=237. 

299.  When  an  equation  is  reduced  by  extracting  an  even 
rbot  of  a  quantity,  the  solution  does  not  determine  whether 
the  answer  is  positive  or  negative.  (Art.  297.)  But  what 
is  thus  left  ambiguous  by  the  algebraic  process,  is  frequently 
settled  by  the  statement  of  the  problem. 

Prob.  3.  A  merchant  gains  in  trade  a  sum,  to  which  320 
dollars  bears  the  same  proportion,  as  five  times  this  sum  does 
to  2500.     What  is  the  amount  gained  ? 

Let  .r=thc  sum  required. 
a  =  320 
6=2500. 


EQUATIONS.  1 3S 

By  the  supposition  a:x::5x  :b 

Multiplying  extremes  and  means        5x^  =al 


vind 


abV 


=(?) 


/320x2500\^ 
Restoring  the  numbers,  .x  =  ^ ;: j   =400. 

Here  the  answer  is  not  marked  as  ambiguous,  because  by 
the  statement  of  the  problem  it  is  gain,  and  not  loss.  It 
must  therefore  be  positive.  This  might  be  determined,  in 
the  present  instance,  even  from  the  algebraic  process. 
Whenever  the  root  of  x^  is  ambiguous,  it  is  because  we  are 
ignorant  whether  the  power  has  been  produced  by  the  mul- 
tiplication of  +x,  or  of  —  oj,  into  itself.  (Art.  262.)  But 
here  we  have  the  multiplication  actually  performed.  By 
turning  back  to  the  two  first  steps  of  the  equation,  we  find 
that  5x^  was  produced  by  multiplying  5x  into  x,  that  is  -f  5r 
into  +a:. 

Prob.  4.  The  distance  to  a  certain  place  is  such,  that  if 
96  be  subtracted  from  the  square  of  the  number  of  miles, 
the  remainder  will  be  48.     What  is  the  distance  ? 

Let  a;=the  distance  required. 
By  the  supposition,  x^  -—96=48 

"  Therefore  x=\/144=12. 

Prob.  5.  If  three  times  the  square  of  a  certain  number 
be  divided  by  four,  and  if  the  quotient  be  diminished  by  12, 
the  remainder  will  be  180.     What  is  the  number? 

By  the  supposition  — r-  —  12=1 80. 


Therefore  a;  =-^256  =  16. 

Prob.  6.  What  number  is  that,  the  fourth  part  of  whose 
square  being  subtracted  from  8,  leaves  a  remainder  equal  to 
four?  Ans.  4. 

Prob.  7.  What  two  numbers  are  those,  whose  sum  is  to 
the  greater  as  10  to  7;  and  whose  sum  multiplied  into  the 
less  produces  270  ? 

Let  10a?= their  sum. 
Then  7a;  =  the  greater,  and  3a?  =  the  less. 
Therefore  a?  =3,  and  the  numbers  required  are  21  and  9. 


L 


134  ALGEBRA. 

Prob.  B»  What  two  numbers  are  those,  whose  difference 
is  to  the  greater  as  2 :  9  ;  and  the  difference  of  whose  squares 
is  128?  Ans.  18  and  14. 

Probi  9-  It  is  required  to  divide  the  number  18  into  two 
iuch  parts,  that  the  squares  of  those  parts  may  be  to  each 
other  as  25  to  IG. 

Let  j^=the  greater  part.  Then  18— A?=the  less. 

By  the  condition  proposed  x^  :  (18— o:)^  : :  25  :  16. 

Therefore  16a;='=25x(l8~a?)2. 

By  evolution  4a: = 5  X  C 1 8  —  a?) . 

And  a;  =  10. 

Prob.  10.  It  is  required  to  divide  the  number  14  into  two 
such  parts,  that  the  quotient  of  the  greater  divided  by  the 
less,  may  be  to  the  quotient  of  the  less  divided  by  the  great- 
er, as  16:9.  Ans.  The  parts  are  8  and  6. 

Prob.  11.  What  two  numbers  are  as  5  to  4,  the  sum  »f 
whose  cubes  is  5103  ? 

Let  5x  and  4a?=the  two  numbers. 

Then  «=3,  and  the  numbers  are  15  and  12.* 

Prob.  12.  Two  travellers  .^  and  J5  set  out  to  meet  each 
©ther,  A  leaving  the  town  C,  at  the  same  time  that  B  left  2>. 
They  travelled  the  direct  road  between  C  and  D ;  and  on 
meeting,  it  appeared  that  ^  had  travelled  18  miles  more 
than  B,  and  that  A  could  have  gone  Bh  distance  in  1 5f  days, 
but  B  would  have  been  28  days  in  going  ./^'s  distance.  Re- 
quired the  distance  between  C  and  Z). 

Let  c€=  the  number  of  miles  A  travelled. 

Then  .t  — 18=  the  number  B  travelled. 

a?-18 
J .  3~  =  Ah  daily  progress. 

a: 
^  =  Bh  daily  progress. 

a?— 18     X 
Therefore  x  :  x—lS  : :  "TTS"'  •  98* 

This  reduced  gives  08=72,  A''s  distance. 

The  whole  distance,  therefore,  from  C  to  Z)=:126  miles. 


EQUATIONS.  135, 

Prob.  1 3.  Find  two  numbers,  which  are  to  each  other  as 
8  to  5,  and  whose  product  is  360.  Ans.  24  and  15. 

Prob.  14.  A  gentleman  bought  two  pieces  of  silk,  whicU 
together  measured  36  yards.  Each  of  them  cost  as  many 
ghiUings  by  the  yard,  as  there  were  yards  in  the  piece,  and 
their  whole  prices  were  as  4  to  1.  What  were  the  lengths 
of  the  pieces  ?  Ans.  24  and  12  yards. 

Prob.  15.  Find  two  numbers,  which  are  to  each  other  as 
3  to  2  ;  and  the  difference  of  whose  fourth  powers,  is  to  the 
sum  of  their  cubes,  as  26  to  7. 

Ans.  The  numbers  are  6  and  4. 

Prob.  16.  Several  gentlemen  made  an  excursion,  each 
taking  the  same  sum  of  money.  Each  had  as  many  servants 
attending  him  as  there  were  gentlemen,  the  number  of  dol- 
lars which  each  had  w  as  double  the  number  of  all  the  ser- 
vants, and  the  whole  sum  of  money  taken  out  was  3456  dol- 
lars.    How  many  gentlemen  were  there  ?  Ans.  12. 

Prob.  1 7.  A  detachment  of  soldiers  from  a  regiment  be- 
ing ordered  to  march  on  a  particular  service^  each  company 
furnished  four  times  as  m.any  men,  as  there  were  companies 
in  the  whole  regiment ;  but  these  being  found  insufficient, 
each  company  famished  3  men  more ;  when  their  number 
was  found  to  be  increased  in  the  ratio  of  17  to  16.  How 
many  companies  were  there  in  the  regiment  ?       Ans.  12. 

AFFECTED  QUADRATIC  EQUATIONS. 

300.  Equations  are  divided  into  classes,  which  are  distin- 
guished from  each  other,  by  the  power  of  the  letter  that  ex- 
presses the  unknown  quantity.  Those  which  contain  only 
the  first  power  of  the  unknown  quantity,  are  called  equa- 
tions of  07ie  dimension^  or  equations  of  the  first  degree. 
Those  in  which  the  highest  power  of  the  unknown  quantity 
is  a  square^  are  called  quadratic^  or  equations  of  the  second 
degree  ;  those  in  which  the  highest  power  is  a  cuhe^  equations 
of  the  third  degree^  fyc» 

Thus  x-=sia-\-h,  is  an  equation  of  the  first  degree. 

x^  =c,  and  x^  -^-ax^d^  are  quadratic  equations,  or  equa- 
tions of  the  second  degree. 

x^  =:h,   and  a?'-f«a;*-f-6a:=J,  are  cubic  equations,  ©r 
equations  of  the  third  degree. 


J36  ALGEBRA. 

301.  Equations  are  also  divided  into  pure  and  affected 
equations.  A  pure  equation  contains  only  one  power  of  the 
unknown  quantity.  This  may  be  the  first,  second,  third,  or 
any  other  power.  An  affected  equation  contains  different 
powers  of  the  unknown  quantity.     Thus, 

V  x^  z=d—h,  is  a  pure  quadratic  equation. 
I  x^  -{-bx^d,  an  affected  quadratic  equation. 
^x^  ~b  —  c,  a  pure  cubic  equation. 
lx^-\-ax^  -\-bx=h,  an  affected  cubic  equation. 

A  pure  equation  is  also  called  a  simple  equation.  But  this 
term  has  been  applied  in  too  vague  a  manner.  By  some 
writers,  it  is  extended  to  pure  equations  of  every  degree  :  by 
others,  it  is  confined  to  those  of  the  first  degree. 

In  a  pure  equation,  all  the  terms  which  contain  the  un- 
known quantity  may  be  united  in  one,  (Art.  185,)  and  the 
equation,  however  complicated  in  other  respects,  may  he. 
reduced  by  the  rules  which  have  already  been  given.  But 
in  an  affected  equation,  as  the  unknown  quantity  is  raised  to 
different  powers,  the  terms  contaming  these  powers  can  not 
be  united.  (Art.  230.)  There  are  particular  rules  for  the 
reduction  of  quadratic,  cubic,  and  biquadratic  equations. 
Of  these,  only  the  first  will  be  considered  at  present. 

302.  An  affected  quadratic  equation  is  oxe  which 

CONTAINS  the  UNKNOWN  QUANTITY  IN  ONE  TERM,  AND  THE 
SQUARE  OF  THAT  QUANTITY  IN  ANOTHER  TERM. 

The  unknown  quantity  may  be  originally  in  several  terms 
of  the  equation.  But  all  these  may  be  reduced  to  two,  one 
containing  the  unknown  quantity,  and  the  other  its  square. 

303.  It  has  already  been  shown  that  a  pure  quadratic  is 
solved  by  extracting  the  root  of  both  sides  of  the  equation. 
An  affected  quadratic  may  be  solved  in  the  same  way,  if  the 
member  which  contains  the  unknown  quantity  is  an  exact 
square.     Thus  the  equation 

x^+2ax-\-a^=:b+h 

may  be  reduced  by  evolution.  For  the  first  member  is  the 
square  of.  a  binoraial  quantity.  (Art.  264.)  And  its  root  is 
:r+a.     Therefore, 

x  4-  a  =  v^6  -f-  A,  and  by  transposing  a, 


QUADRATIC  EQUATIONS.  j  37 

304.  But  it  is  not  often  the  case,  that  a  member  of  an  af- 
fected quadratic  equation  is  an  exact  square,  till  an  addition- 
al term  is  applied,  for  the  purpose  of  making  the  required 
reduction.     In  the  equation 

the  side  containing  the  unknown  quantity  is  not  a  complete 
square.  The  two  terms  of  which  it  is  composed  are  indeed 
such,  as  might  belong  to  the  square  of  a  binomial  quantity. 
(Art.  214.)  But  one  term  is  xoctTifm^.  We  have  then  to  in- 
quire, in  what  way  this  may  be  supplied.  From  having  tzvo 
terms  of  the  square  of  a  binomial  given,  how  shall  we  find 
the  third  ? 

Of  the  three  terms,  two  are  complete  powers,  and  the 
other  is  twice  the  product  of  the  roots  of  these  powers ; 
(Art.  214,)  or,  which  is  the  same  thing,  the  product  of  one 
of  the  roots  into  twice  the  other.     In  the  expression 

the  term  ^ax  consists  of  the  factors  2«  and  x*  The  latter  13 
the  unknown  quantity.  The  other  factor  2a  may  be  consid- 
ered the  co-efficient  of  the  unknown  quantity ;  a  co-efficient 
being  another  name  for  a  face  tor,  (Art.  41.)  As  a?  is  the 
root  of  the  first  term  x^  ;  the  other  factor  2a  is  twice  the 
root  of  the  third  term,  which  is  wanted  to  complete  the 
square.  Therefore  half  2a  is  the  root  of  the  deficient  term, 
aiid  ft*  is  the  term  itself.  The  square  completed  is 
a?^+2aa;  +  «*, 

where  it  will  be  seen  that  the  last  term  a*  is  the  square  of 
half  2fif,  and  2a  is  the  co-efficient  of  x  the  root  of  the  first 
term. 

In  the  same  manner,  it  may  be  proved,  that  the  last  term 
of  the  square  of  any  binomial  quantity,  is  equal  to  the 
square  of  half  the  co-efficient  of  the  root  of  the  first  term. 
From  this  principle,  is  derived  the  following  rule : 

305.  To  COMPLETE  THE  SQUARE,  in  an  affected  quadratic 
equation ;  take  the  sc^uare  or  half  the  co-EFnciENT  of 

THE  first  POWER  OF  THE  UNKNOWN  QUANTITY,  AND  ADD  IT  TQ 
BOTH  SIDES  OF  THE  EQUATION. 

Before  completing  the  square,  the  known  and  unknown 
quantities  must  be  brought  on  opposite  sides  of  the  equation 
by  transposition;  and  the  highest  power  of  th«  unknowa 
19 


238  ALGEBRA. 

quantity  must  have  the  affirmative  sign,  and  be  cleared  of 
fractions,  co-efficients,  &;c.     See  arts.  308,  9,  10,  11. 

^i!er  the  square  is  completed,  the  equation  is  reduced,  by 
extracting  the  square  root  of  both  sides,  and  transposing  the 
known  part  of  the  binomial  root.     (Art.  303.) 

The  quantity  which  is  added  to  one  side  of  the  equation, 
to  complete  the  square,  must  ,be  added  to  the  other  side  al- 
so, to  preserve  the  equality  of  the  two  members.    (Ax.  1.) 

306.  It  will  be  important  for  the  learner  to  distinguish  be- 
tween what  is  peculiar  in  the  redaction  of  quadratic  equa- 
tions, and  what  is  common  to  this  and  the  other  kinds  which 
have  already  been  considered.  The  peculiar  part,  in  the 
resolution  of  affi^cted  quadratics,  is  the  completing  of  the 
square.  The  other  steps  are  similar  to  those  by  which  pure 
equations  are  reduced. 

For  the  purpose  of  rendering  the  completing  of  the 
square  familiar,  there  will  be  an  advantage  in  beginning  with 
examples  in  which  the  equation  is  already  prepared  for  this 
step. 

Ex.  1-  Reduce  the  equation      ^        x^ ■\-Qaw=.b 

Completing  the  square  a?*  +6aa7-f9a^  =9a^  +6 

Extracting  both  sides    (Art.  303.)  x-^Za^t^^a^ +h 

And  a:  =  -3o±\/9aM^ 

Here  the  co-efficient  of  x^  in  the  first  step,  is  6a ; 

The  square  of  half  this  is  9a^ ,  which  being  added  to  both 
sides  completes  the  square.  The  equation  is  then  reduced 
by  extracting  the  root  of  each  member,  in  the  same  manner 
as  in  art.  297,  excepting  that  the  square  here  being  that  of 
'a  binomial,  its  root  is  found  by  the  rule  in  art.  265. 

2.  Reduce  the  equation  x^ —Sbx—h 

Completing  the  square^  sv^  —Qbx-{-  16b^  =  \6b2  -^-h 

Extracting  both  sides  X'-4b=lV I6b^  -{-h 
And  x=4b±VlGb^-{-h 

In  this  example,  half  the  co-efficient  of  x  is  4ft,  the  square 
of  which  166*  is  to  be  added  to  both  sides  of  the  equa- 
tion. 


QUADRATIC  EQUATIONS,  139 

3.  Reduce  the  equation  x^-\-ax=b-\-b 


a^     a- 


Completing  the  square  x^+ax-i-— =—  -{-b+h 

By'evolution  ^  -f  —  =  _  (^— +b-{-hj 


And 


«+/«* 


.=-|l(^+*+0^- 


4.  Reduce  the  equation  x^—x—h—d 
Completing  the  square,          a;2--a?4-4=i  +  ^-- ^ 
And                                          a;=il(i+A-A) 

Here  the  co-efficient  of  ^  is  1 ,  the  square  of  half  which  is  J. 

5.  Reduce  the  equation  a:*+3a;=J+6 
Completing  the  square,          a?*  +  3a?-ff  =1+^+6 

And  ;^=-|±(f4-rf+6f, 

6.  Reduce  the  equation     x^ —abx=-ah—cd 

Completing  the  square,  x^  —  a&a;  +  — j— =--  — +«fo—  ea 
And  a?=Y__("~^  +  «i— crfj 

7.  Reduce  the  equation     x^-\--^^h 

ax      a2       fl2 
Completing  the  square,  ^^  +y +^=462+^ 

a  I..  /  a^        \i 
And  ^=-26-146^"^^;   • 

By  art  1 58,  y  =  y  X  a?.     The  co-efficient  of  a;,  therefore, 

a  a 

T-.    Half  of  this  is  ^. 


is  -r.    Half  of  this  is  ^,  (Art.  163.)  the  square  of  which  is 


a' 
4b^* 


140  ALGEBRA. 

oc 
8.  Reduce  the  equation    x^—-r^'7h» 

X        1  1 

Completing  the  square,  a:^  --y + JJi  =^2 +'^^^ 

And  ^=26-(46^  +  ^^0'- 

X        1 

Here  the  fraction  T  —  ~^X^*  (Art.  158.)     Therefore  the 

co-efficient  of  :v  is  y^, 

307.  Tn  these  and  similar  instances,  the  root  of  the  third, 
term  of  the  completed  square  is  easily  found,  because, this 
root  is  the  same  half  co-efficient  from  which  the  term  has 
just  been  derived,     (Art.  304.)     Thus  in  the  last  example, 

half  the  co-efficient   of    x  is   qT,  and  this  is  the  root  of  the 

1 

third  term  tt^. 

308.  When  the  first  power  of  the  unknown  quantity  is  in 
stveral  terms,  these  should  be  united  in  one,  if  they  can  be 
by  the  rules  for  reduction  in  addition.  But  if  there  are  lit- 
eral co-efficients,  these  may  be  considered  as  constituting, 
together,  a  compound  co-efficient  or  factor,  into  which  the 
unknown  quantity,  is  multiplied. 

Thus  ax+bx+dx^^ia  +  b  +  d^Xx.  (Art.  120.)  The 
square  of  half  this  compound  co-efficient  is  to  be  added  to 
"both  sides  of  the  equation. 

1 ,  Reduce  the  equation  x^-{-Sx+2x-\' x=d 
Uniting  terms,  x^-{-6x=d 
Completing  the  sqtiare,  a:^  +  6a? -|- 9 =94- d 
And  x=''3±^/9+d 

2.  Reduce  the  equation       ac^  +  ax+bx^k 

By  art.  120,  x^  +  {a+b)xx=k 

/a-^by      /a  +  bV 
Therefore      ^^^ +  («+&)  X  at-]- ^-^j  ==\-Y')   +^ 

a+h     +    l/a+b\''     ^ 
By  evolution,  isp+^-Y""  — v  V'~2~/  "^ 


,    ,  a+b+    ira  +  by 

And  A'=— •— 7 


/^)-« 


QUADRATIC  EQUATIONg.  1 4 1 

3.  Reduce  the  equation    x^'\-ax—x=b 
Byart.  120  x2-{-{a'-'i)xx=h 

Therefore  ^^  +  {(t'-l)xx+{-^-  J  =\-^~)   +h 


And 


a  — 1+    //a— 1\^ 


309.  After  becoming  familiar  with  the  method  of  comple- 
ting the  square,  in  affected  quadratic  equations,  it  will  be 
proper  to  attend  to  the  steps  Avhich  are  preparatory  to  this. 
Here,  however,  httle  more  is  necessary,  than  an' application 
of  rules  already  given.  The  known  and  unknown  quanti- 
ties must  be  brought  on  opposite  sides  of  the  equation  by 
transposition.  And  it  will  generally  be  expedient  to  make 
the  square  of  the  unknown  quantity  the  first  or  leading  term, 
as  in  the  preceding  examples.  This  indeed  is  not  essential. 
But  it  will  show,  to  the  best  advantage,  the  arrangement  of 
the  terms  in  the  completed  square^, 

1.  Reduce  the  equation  a-^-dx — 3^=3;^— atz 
Transp.  and  uniting  terms  x^-^^x^Sb—a 
Completing  the  square  :v2  -f  2jc  -j- 1=  1  -|-  3 J — « 
And  x=-^l±VT+3b^a. 

X       36 

2.  Reduce  the  equation  9~'~X2~^ 

Clearing  of  fractions,  &c.     :v2-[-10a:=56 

Completing  the  square         a;^  -+-1 0^^+25 =25 +  56= 81 

And  Ar==-5±v/81  =  -5±9. 

310.  If  the  highest  power  of  the  unknown  quantity  has  anj 
co-efficient  or  divisor,  it  must,  before  the  square  is  completed, 
by  the  rule  ifi  art.  305,  be  freed  from  these,  by  multiplica- 
tion or  division,  as  in  arts.  180  and  184. 

Reduce  the  equation  x^-\'2iia—Qh=12x^5x^ 

Transp.  and  uniting  terms j^  6;^-^  —  1 2^=6/i — 24a 

Dividing  by  6,  x^  —2x=h—  4a 

Completing  the  square,  x^'-'^x-{-l^l^h—4(t 
Extracting  and  transp .  x=;:l ± V 1+^— 4a 


J  42  ALGEBRA. 


2.  Reduce  the  equation     h-\-2x=d—^- 


a 


Clearing  of  fractions      hx^ -{-^ax^^ad-ah 

■p..   .,.      ,      ,  „      ^ax     ad—ah 

l>ividing  by  h,  x^  ■\—r  == — i 

o  b 

rr^y        .  ^ax     a2     ^2      ad —ah 

Therefore  ;..+_-+^=-_+_-^_ 

5n.  If  the  square  of  the  unknown  quantity  is  in  several 
terms,  ihe  equation  must  be  divided  by  all  the  co-efficients 
of  this  square,  as  in  art.  185. 

1 .  Reduce  the  equation  bx^ -{-dx^  —4x:=b—h 

Dividing  hy  b+d,  (Art.  121.)^^       ^"^      ^ "  ^ 


b+d~b+d 


Therefore 


^—b-\-d-y^b4-d/  + 


i-A 


2.  Reduce  the  equation  «x2  +  x=rA-|-3a7  — o.^ 

Transp.  and  uniting  terms         ax^  -{-x^  —2x=h 

2^  h 

Dividing  by  «+ 1,  ,,._-_=.-_ 

Completing  the  square  ^^-^grj+C^;  -(^l)  +^1 

TSxtracting  and  transp.    ^=~t-\l{~)  +^^^ 

There  is  another  method  of  completing  the  square,  which, 
in  many  cases,  particularly  those  in  which  the  highest  pow- 
er of  the  unknown  quantity  has  a  co-efficient,  is  more  sim*' 
pie  in  its  application,  than  that  given  in  art.  305. 
Let  ax^  -^-bx—d. 

If  the  equation  b6  multiplied  by  4a,  and  if  6^  be  added 
to  both  sides,  it  will  become 

Aa^x^'+Aabx^b^^Ud^b''  ; 
the  first  member  of  which  is  a  complete  power  of  ^ax-\-b. 

Hence, 

311.&.  In  a  quaJratic  equation,  the  square  may  be  com- 
pleted, by  multiplying  the  equation  into  4  times  the  co-^ffi- 


QUADRATIC  EQUATIONS.  14^'^ 

eient  of  the  highest  power  of  the  unknown  quantity,  and 
adding  to  hoth  sides,  the  square  of  the  co-efficient  of  the 
lowest  power. 

The  advantage  of  this  method  is,  that  it  avoids  the  intro- 
duction of  fractions^  in  completing  the  square. 

This  will  be  seen,  by  solving  an  equation  by  both  methods. 

Let  ax^  ^dx=^h> 

Completing  the  square,  by  the  rule  just  given  ; 

4a^^*  -f4aJa7-f- J«  =  4«A+(^^ 

Extracting  the  root  2q.v-\-d=±y/^ah-\-d'^ 


—d±y/^ah+d'' 
And  x= — 


2a 

Completing  the  square  of  the  given  equation  by  arts.  305 

dx      d"-       h      d^ 
and  310; 


Extracting  the  root 


^'+a+4a^- 

a'^Aa'' 

d      +    Ih 

d^ 

d+    Ih 

And 

If  a==  1,  the  rule  will  be  reduced  to  this  :  "  Multiply  th^ 
equation  by  4,  and  add  to  both  sides,  the  square  of  the  co- 
efficient of  or." 

Letx''+dx=h 

Completing  the  square  4x^  +  ^dx  -{-d^-=  4/i -f  d^ 

Extracting  the  root  2x + d=^±V4h+dF 

-dlV/lh-^-d^ 
And  ^= ^ . 

1 .  Reduce  the  equation  3^^  -f  5x  =42 
Completing  the  square  36x^  -f  GOor-f  25=529 
Therefore  x—3, 

2.  Reduce  the  equation  x^  —]5x=--54;  ^ 
Completing  the  square  4^:^ —60;c-f  225=9  ' 
Therefore  2.r  =  1513=  1 8  or  1 2. 

312.  In  the  square  of  a  binomial,  the  first  and  last  terms 
are  always  positive.     For  each  is  the  square  of  one  of  the 


144  ALGEBRA. 

terms  of  the  root,  (Art.  214.)  But  every  square  is  positive. 
(Art.  218.)  If  then  —x^  occurs  in  an  equation,  it  can  not, 
with  this  sign,  form  a  part  of  the  square  of  a  binomial.  But 
if  all  the  signs  in  the  equation  be  changed,  the  equality  of 
the  sides  will  be  preserved,  (Art.  177,)  the  term  —x^  wilt 
become  positive,  and  the  square  may  be  completed. 

1.  Reduce  the  equation  ^x^  ■\.^x=id—li 
Changing  all  the  signs  .  ^r^  —  2x=^--c? 
Therefore  ^ = 1±  Vl+A— t/, 

2.  Reduce  the  equation  4^— :c^  =  — 12 

Answer  x—^lyj\^. 

313.  In  a  quadratic  equation,  the  first  term  or^  is  the 
square  of  a  single  letter.  But  a  binomial  quantity  may  con- 
sist of  terms,  one  or  both  of  which  are  already  powers. 

Thus  a:  ^ -fa  is  a  binomial,  and  its  square  is 

^^  +  2a.r3-|-a2^  ^ 

where  the  index  of  x  in  the  fifst  term  is  twice  as  great  as  in 
the  second.  When  the  third  term  is  deficient,  the  square 
may  be  completed  in  the  same  manner  as  that  of  any  other 
binomial.  For  the  middle  term  is  twice  the  product  of  the 
roots  of  the  two  others. 

So  the  square  of  ^"-fa,  is  ^*n-f2aar**-|-a'. 

And  the  square  of  x^  -fa,  is  x«  -f  2aj?"  -fa*. 
Therefore, 

3H.  Any  equation  which  contains  only  two  differ- 
ent POWERS  or  roots  OF  THE  UNKNOWN  QUANTITY,  THE  IN- 
DEX OF  ONE  OF  WHICH  IS  TWICE  THAT  OF  THE  OTHER,  MAY  BE 
resolved  IN  THE  SAME  MANNER  AS  A  QUADRATIC  EQUATION^ 
BY  COMPLETING  THE  SQUARE. 

It  must  be  observed,  however,  that  in  the  binomial  root, 
the  letter  expressing  the  unknown  quantity  may  still  have  a 
fractional  or  integral  index,  so  that  a  farther  extraction,  ac- 
cording to  art.  297,  may  be  necessary. 

1.  Reduce  the  equation  x'^^x^=zh—a 

Completing  the  square  cc*—  :r^-fT  =  J+&— c? 

Extracting  and  transp.  ^*  =  |1.  Vl  -f  6 — a 


Extracting  again,  (Art,  297,)      .t==±  ^ ^i ^^i+6-a. 


QUADRATIC  EQUATIONS.  145 

2.  Reduce  the  equation  a:^»— 46a;"=a 

«  * — ' ', 

Answer         x=:t'^  2h±\/4b^'{a, 

3.  Reduce  the  equation  «+4v'a?=A— n 
Completing  the  square  a?+4'/a?4-4=^— ?i-f4 
Extracting  and  transp.             ^x=:  —21.^ h—n-\- 4 


Involving  ic={— 2lV/i— n+4)^ 

a  i 

4.  Reduce  the  equation,  xn  +8a;"=ra+& 

1         i 
Competing  the  square  aj"  +  8a:"  + 1 6 = a  4  ^4 1 6 


i 


Extracting  and  transp.  cc^=z—4ly/a+b  +  lQ 

Involving  .     a:=(-4l\^a4H^)'^. 

315.  The  solution  of  a  quadratic  equation,  whether  pure 
or  affectedj  gives  two  results.  For  after  the  equation  is  re- 
duced, it  contains  an  ambiguous  root.  In  a  pure  quadratic, 
ihis  root  is  the  whole  value  of  the  unknown  quantity.  (Art. 
297.) 

Thus  the  equation  ac2  =64 

Becomes,  when  reduced,  a=l-v/64 

That  is,  the  value  of  x  is  either  4-8  or  —8,  for  each  of 
these  is  a  root  of  64.  Here  both  the  values  of  x  are  the 
same,  except  that  they  have  contrary  signs.  This  will  be 
the  case  in  every  pure  quadratic  equation,  because  the  whole 
of  the  second  member  is  under  the  radical  sign.  The  two 
values  of  the  unknown  quantity  will  be  alike,  except  that 
one  will  be  positive,  and  the  other  negative. 

316.  But  in  affected  quadratics,  apart  only  of  one  side  of 
the  reduced  equation  is  under  the  radical  sign.  When  this 
part  is  added  to,  or  subtracted  from,  that  which  is  without 
the  radical  sign  ;  the  two  results  will  differ  in  quantity,  and 
will  have  their  signs  in  some  cases  alike,  and  in  others  un- 
like. 

1.  The  equation  oC^  48^=20 

Becomes,  when  reduced  ac  =  —  4l  V 1 6  4  20. 

That  is  ;c=  — 4±6. 

Here  the  first  value  of  » is,  —  4  4  6  =  4- 2  )   one  positive,  and 
And  the  second  is^  —4—6=.  — 10  J  the  other  negative. 

20 


146  ALGEBRA. 

2.  The  equation  a:^--8a;  =  — 15 

Becomes,  when  reduced     a;=4l'Vl6  — 15 
That  is  a:=4±l 

Here  the  first  vahie  of  a?  is  4+1  =  +  5  )  ,    .,         ... 
And  the  second  is  4- 1  =  +  3  T       ^         ''' 

That  these  two  values  of  x  are  correctly  found,  may  he 
proved,  by  Substituting  first  one,  and  then  the  other,  for  oc 
itself,  in  the  original  equation.     (Art.  194.) 

Thus52~8><5  =  25-40=-15 
And    3^ -8x3=^9-24= -15. 

317.  In  the  reduction  of  an  affected  quadratic  equation^ 
the  value  of  the  unknown  quantity  is  frequently  found  to  be 
imaginary. 

Thus  the  equation  x^—Sx  =  —^0 

Becomes,  when  reduced,  a?  =141^1 6— 20 

That  is,  a;=4±V-4. 

Here  the  root  of  the  negative  quantity  —4  can  not  be  as- 
signed, (Art.  263,)  and  therefore  the  value  of  x  can  not  be 
found.  There  *will  be  the  same  impossibility,  in  every  in- 
stance in  which  the  negative  part  of  the  quantities  under  the 
radical  sign  is  greater  than  the  positive  part.* 

318.  Whenever  one  of  the  values  of  the  unknown  quanti- 
ty, in  a  quadratic  equation,  is  imaginary,  the  other  is  so  also. 
For  both  are  equally  affected  by  the  imaginary  root. 

Thus,  in  the  example  above, 

The  first  value  of  a;  is  4+  ■\/"~4, 

And  the  second  is  4—  -v/  — 4  ;  each  of  which 
contains  the  imaginary  quantity  V— 4. 

319.  An  equation  which  when  reduced  contains  an  ima- 
ginary root,  is  often  of  use,  to  enable  us  to  determine  wheth- 
er a  proposed  question  admits  of  an  answer,  or  involves  an 
absurdity. 

Suppose  it  is  required  to  divide  8  into  two  such  parts,  that 
the  product  will  be  20. 

*  See  Note  G-* 


QUADRATIC  EQUATIONS.  I47 

If  X  is  one  of  the  parts,  the  other  will  be  8— a;.  (Art.  195.) 
By  the  conditions  proposed  (8  —x)  X  x =20 

This  becomes,  when  reduced,  a;=4i\/--4. 

Here  the  imaginary  expression  -/— 4  shows  that  an  an- 
swer is  impossible  ;  and  that  there  is  an  absurdity  in  suppo- 
sing that  8  may  be  divided  into  two  such  parts,  that  their 
product  shall  be  20. 

320.  Although  a  quadratic  equation  has  two  solutions,  yet 
both  these  may  not  always  be  applicable  to  the  subject  pro- 
posed. The  quantity  under  the  radical  sign  may  be  produ- 
ced either  from  a  positive  or  a  negative  root.  But  both 
these  roots  may  not,  in  every  instance,  belong  to  the  prob- 
lem to  be  solved.     See  art.  299. 

Divide  the  number  30  into  two  such  parts,  that  their  pro- 
duct may  be  equal  to  8  times  their  difference. 

If  a;=the  lesser  part,  then  30— a;  =  the  greater. 

By  the  supposition,     ;^  X  (30 — a;)  =  8  x  (30 — 2a?) 

This  reduced,  gives    a; = 2311 7 =40  or  6= the  lesser  part. 

But  as  40  can  not  be  a  part  of  30,  the  problem  can  have 
but  one  real  solution,  making  the  lesser  part  6,  and  the 
greater  part  24. 

m'  Examples  of  Quadratic  Equations. 

1.  Reduce  3a;  ^— 9a?— 4  =  80.  Ans.  a?=7,  or— 4. 

36— a; 

2.  Reduce  4a;  — =46.  Ans.  a;  =  12,  or— |. 

a;  ■* 

14— a; 

3.  Reduce  4x—  ^-37^-= 14.  Ans.  a;=^4,  or— f. 

3a:-3  3Af-6 

4.  Reduce  5a;— — —^^rz^x-^-     ^ — '      Ans.  a; =4,  or— 1. 

16     ]00-9a; 

5.  Reduce— -— — —5— =3.  Ans.  a;=:4,  or  2-fV 

3a;  — 4  a;— 2 

6.  Reduce —^y-f- 1  =  10  — —^*         Ans.  a; =12^  or  6. 

a?4-4     7— a;     4a?4-7 

7.  Reduce -^—^3^=—^ — —1.     Ans.  a; =21,  or  5. 


148                                          ALGEBKA. 

8.  Reduce  ^3  «  6^   +9-^-3- 

Ans.  a;  =  ljOr— 28. 

9.  Reduce  ^^  J +^=3. 

Ans.  a: =2. 

3a;       a:- 1 

Ans.  a?  =10. 

10.  Keauce^_^2""    6    ~'^'""^- 

X     a     2 

11.  Reduce -+-=-• 

Ans.  aj^lt-Zl-a^. 

12.  Reduce  a;*+ax*=6.  Ans.:tf=^— -^— y^-f — j 

a;«     a?3           1  n 

13.  Reduce  2--—=  —  ^.  Ans.  a;=y^. 

2  1 

14.  Reduce  2aj^+3x"^=2.  Ans.  x=:il. 

15.  Reduce  ^.T—i  V'3?=22i.  Ans.  a?=49. 

16.  Reduce  2a;*— a; « +96 =99.  Ans.  x=l^e. 

17.  Reduce  (10+a;)^-(10-|-a;)^=2.       Ans*  a;=6. 

18.  Reduce  3a?^»»— 2a;«=8.  Ans.  a;=^2. 


2h 


19.  Reduce  2(1 +a;-.a;2)-  Vl4.a?-«2  =  -i/  ^; 

Ans.  a;=|+Jv'9^ 

,   b  +    I4a^-b^ 

20.  Reduce  V'a?^—a^=a;  — 6.  Am.x  =  -a^\l — r 

V4a;  +  2     4— /a: 

21.  Reduce  7^:77=— 7^-.  Ans.  a:=4. 

6  3 

22.  Reduce  x^-{-x^=156.  Ans.  a;=243. 

, 21 

23.  Reduce  V 2a; 4- 1+2-/^"  =-7==".     Ans.  a; =4. 

v2a;+l 

7a-\-5x 


24.  Reduce  2\^a;— a-f  3v^2x=7=r-.    Ans.  a:=9a. 

vx—a 

25.  Reduce  a?+16~7v'a?4-16  =  10--4v'a;  +  16.     Ans.  a;=9. 

26.  Reduce  Va;*+\/a3*=6-/a:. 

Dividing  by  v/a?,         x^+x=s6        Anda;=2* 


QUADRATIC  EQUATION 

S.                        149 

4a;~5     3a;--7     9^+23 
27.  Reduce     ^        3^^^-     ^3^    . 

Ans.  a:=2. 

3                6            11 

Ans.  a:  =  3. 

28.  Reduce  g^_^.  +;^*  +  2a:-5^' 

29.  Reduce  (a;~5)'-3(a?-5)^=40. 

Ans.  a?=:9. 

30.  Reduce   x4'Vx+6=2+3Vx+G. 

Ans.  cr^lO. 

PROBLEMS  PRODUCING  QUADRATIC  EQUATIONS. 

Prob.  1.  A  merchant  has  a  piece  of  cotton  cloth,  'and  a 
piece  [of  silk.  The  number  of  yards  in  both  is  110;  and  if 
the  -square  of  the  number  of  yards  of  silk  be  subtracted 
from  80  times  the  number  of  yards  of  cotton,  the  difference, 
will  be  400.     How  many  yards  are  there  in  each  piece  ? 

Let  a?=the  yards  of  silk. 
Then  110— a;  =  the  yards  of  cotton. 
By  supposition         400  =  80X(110— a;J— ar^ 
Therefore  ^=-40-v/i  0000= -401100.       1., 

The  first  value  of  x,  is  —40+100=60,  the  yards  of  silk ; 
And  110— jp=  110— 60=50,  the  yards  of  cotton. 

The  second  value  of  x,  is  —40—100=^  —  140;  but  as  this 
is  a  negative  quantity,  it  is  not  applicable  to  goods  which  a 
man  has  in  his  possession. 

Prob.  2.  The  ages  of  two  brothers  are  such,  that  their 
sum  is  45  years,  and  their  product  500.  "What  is  the  age  of 
each  ?  Ans.  25  and  20  years. 

Prob.  3.  To  find  two  numbers  such,  that  their  difference 
shall  be  4,  and  their  product  117. 

Let  x=one  number,  and  07+4  =  the  other. 
By  the  conditions  (:r-j-4)x^=117 

This  reduced,  gives  j:  =  —  ^iV  1 2 1  =  —  21 1 1  • 

One  of  the  numbers  therefore  is  9,  and  the  other  13. 

Prob.  4.  A  merchant  having  sold  a  piece  of  cloth  which 
cost  him  30  dollars,  found  that  if  the  price  for  which  he  sold 
it  were  multipHed  by  his  gain,  the  product  would  be  equal 
to  the  cube  of  his  gain.     What  was  his  gain  ? 


150  ALGEBRA.  ^ 

Let  a;=the  gain. 
Then  30  4-^= the  price  for  which  the  cloth  was  sold. 
By  the  statement  x^={30+^)x^ 

Therefore  a:=iVi  +  30=l±^^ 

The  first  value  of  xis  |4-Y  =  -f-6. 
The  second  value  is      ^— -/ 


+  6.  > 
-5.  5 


As  the  last  answer  is  negative,  it  is  to  be  rejected  as  in- 
♦  onsistent  with  the  nature  of  the  problem,  (Art.  320,)  for 
gain  must  be  considered  positive. 

Prob.  5.  To  find  two  numbers,  whose  difference  shall  be 
3,  and  the  difference  of  their  cubes  1 1 7. 

Let  :r=the  least  number. 
Then  x-{-3  =z  the  greatest. 
By  supposition  (^  +  3)  '  —^  ^  =  117 

Expanding  (a; +  3) 3 (Art.  217.)  9:^2 +27:c=  117-^27=90 
And  :r=-|±  Vv  =  -|±i. 

The  two  numbers,  therefore,  are  2  and  5. 

Prob.  6.  To  find  two  numbers,  whose  difference  shall  be 
12,  and  the  sum  of  their  squares  1424. 

Ans.  The  numbers  are  20  and  32. 

Prob.  7.  Two  persons  draw  prizes  in  a  lottery,  the  differ- 
ence of  which  is  120  dollars,  and  the  greater  is  to  the  less, 
n^s  the  less  to  10.     What  are  the  prizes  ? 

Ans.  40  and  160. 

Prob.  8.  What  two  numbers  are  those  whose  sum  is  6, 
and  the  sum  of  their  cubes  72  ?  Ans".  2  and  4. 

Prob.  9.  Divide  dhe  number  56  into  two  such  parts,  that 
their  product  shall  be  640. 

Putting  X  for  one  of  the  parts,  we  have,  ^=28112=40  or 
16. 

In  this  case,  the  two  values  of  the  unknown  quantity,  are 
the  two  parts  into  which  the  given  number  was  required  to 
be  divided. 

Prob.  10.  A  gentleman  bought  a  number  of  pieces  of 
cloth  for  675  dollars,  which  he  sold  again  at  48  dollars  by 
the  piece,  and  gained  by  the  bargain  as  much  as  one  piece 
cost  him.     What  was  the  number  of  pieces  ?         Ans,  15. 


QUADRATIC  EQUATIONS.  131 

Prob.  11.^  and  B  started  together,  for  a  place  1'50  miles 
distant.  A'^s  hourly  progress  was  3  miles  more  than  B's, 
and  he  arrived  at  his  journey's  end,  8  hours  and  20  minuter 
before  B.     What  was  the  hourly  progress  of  each  ? 

Ans.  9  and  6  miles. 

Prob.  12.  The  difference  of  two  numbers  is  6  ;  and  if 
47  be  added  to  twice  the  square  of  the  less,  it  will  be  equal 
to  the  square  of  the  greater.     What  are  the  numbers  ? 

Ans.  17  and  11. 

Prob.  13.  ^  and  B  distributed  1200  dollars  each,  among 
a  certain  number  of  persons.  A  relieved  40  persons  more 
than  B,  and  B  gave  to  each  individual  5  dollars  more  than 
A,     How  many  were  reUeved  by  A  and  B  ? 

Ans.  120  by  A,  and  80  by  B. 

Prob.  14.  Find  two  numbers,  whose  sum  is  10,  and  the 
sum  of  their  squares  58  ?  Ans-  7  and  3. 

Prob.  15.  Sevei-al  gentlemen  made  a  purchase  in  compa- 
ny for  1 75  dollars.  Two  of  them  having  withdrawn,  the- 
bill  was  paid  by  the  others,  each  furnishing  10  dollars  more, 
than  would  have  been  his  equal  share,  if  the  bill  had  bee» 
paid  by  the  whole  company,  TVliat  was  the  number  in  the 
company  at  first  ?  Ans.  7. 

Prob.  16.  A  merchant  bought  several  yards  of  linen  for 
60  dollars,  out  of  which  he  reserved  15  yards,  and  sold  the 
remainder  for  54  dollars,  gaining  10  cents  a  yard.  How  ma- 
ny yards  did  he  buy,  and  at  what  price  ? 

Ans.  75  yards,  at  80  cejats  a  yard, 

Prob.  Xl.  A  and  B  set  out  from  two  towns,  which  were 
247  miles  distant,  and  travelled  the  direct  road  till  they  met. 
A  went  9  miles  a  day  ;  and  the  number  of  days  which  thej 
travelled  before  meeting,  was  greater  by  3,  than  the  number- 
of  miles  which  B  went  in  a  day.  How  many  miles  did  each 
travel  ?  Ans.  A  went  117,  and  B  1 30  miles. 

Prob.  18.  A  gentlemen  bought  two  pieces  of  cloth,  the 
finer  of  which  cost  4  shillings  a  yard  more  than  the  other. 
The  finer  piece  cost  £18  ;  but  the  coarser  one,  which  wais 
2  yards  longer  than  the  finer,  cost  only  £16.  How  many 
yards  were  there  in  each  piece,  and  what  was  the  price  of  a 
yard  of  each  ? 

Ans.  There  were  18  yards  of  the  finer  piece,and  20  of  the 
coarsQf ;  and  the  prices  were  20  and  10  shilling?* 


J  52  ALGEBRA. 

Prob.  1 9.  A  merchant  bought  54  gallons  of  Madeira  wine^ 
and  a  certain  quantity  of  Tenerilfe.  For  the  former,  he 
g^ve  half  as  many  shillings  by  the  gallon,  as  there  were  gal- 
lons of  Teneriffe,  and  for  the  latter,  4  shillings  less  by  the 
gallon.  He  sold  the  mixture  at  10  shillings  by  the  gallon, 
and  lost  £28  166.  by  his  bargain.  Required  the  price  of 
the  Madeira,  and  the  number  of  gallons  of  Teneriffe. 

Ans.  The  Madeira  cost  18  shillings  a  gallon,  and  there 
were  36  gallons  of  Teneriffe. 

Prob  20.  If  the  square  of  a  certain  number  be  taken  from 
40,  and  the  square  root  of  this  difference  be  increased  by 
10,  and  the  sum  be  multiplied  by  2,  and  the  product  divided 
by  the  number  itself,  the  quotient  will  be  4.  What  is  the 
number  ?  Ans.  6. 

Prob.  21.  A  person  being  asked  his  age,  replied.  If  you 
add  the  square  root  of  it  to  half  of  it,  and  subtract  12,  the 
remainder  will  be  nothing.     What  was  his  age  ? 

Ans.  16  years. 

Prob.  22.  Two  casks  of  wine  were  purchased  for  58  dol- 
lars, one  of  which  contained  5  gallons  more  than  the  other, 
and  the  price  by  the  gallon,  was  2  dollars  less,  than  ^  of  the 
number  of  gallons  in  the  smaller  cask.  Required  the  num- 
ber of  gallons  in  each,  and  the  price  by  the  gallon. 

Ans,  The  numbers  were  12  and  17,  and  the  price  by  the 
gallon  2  dollars. 

Prob.  23.  In  a  parcel  which  contains  24  coins  of  silver 
and  copper,  each  silver  coin  is  worth  as  many  cents  as  there 
are  copper  coins,  and  each  copper  coin  is  worth  as  many 
cents  as  there  are  silver  coins  ;  and  the  whole  are  worth  Z 
dollars  and  16  cents.     How  many  are  there  of  each  ? 

Ans.  6  of  one,  and  18  of  the  other. 

Prob.  24.  A  person  bought  a  certain  number  of  oxen  for 
80  guineas.  If  he  had  received  4  more  oxen  for  the  same 
money,  he  would  have  paid  one  guinea  less  for  each.  What 
was  the  number  of  oxen  ?  Ans.  16. 

SUBSTITUTION. 

321.  In  the  reduction  of  Quadratic  Equations,  as  well  as 
in  other  parts  of  algebra,  a  complicated  process  may  be  ren- 
dered much  more  simple,  by  introducing  a  new  letter  which 


QUADRATIC  EQUATIONS.  I53 

shall  be  made  to  represent  several  others.  This  is  termed 
substitution,  A  letter  may  be  put  for  a  compound  quantity 
as  well  as  for  a  single  number.     Thus  in  the  equation 

we  may  substitute  6,  for  |+  -v/86~-644-^»  The  equation 
will  then  become         ;p*  --^a^r—b,  and  when  reduced 

will  be  sc:=:atVa^+b. 

After  the  operation  is  completed,  the  compound  quantity 
for  which  a  single  letter  has  been  substituted,  may  be  restor- 
ed. The  last  equation,  by  restoring  the  value  of  6,  will 
become 

:t«=a±Va2-f  3^^86-644-A. 
Reduce  the  equation  a^r— 2x— J=&a7— x^— -x 

Transposing,  &:c.  x'^ +(a—b  —  l)xx=d 

Substituting  h  for  {a —  h —  I),  x^+hx=:d 

Therefore  :^=:-y+VX  +  ^ 

«— &  — 14.    lia-b-\Y~ 
Restoring  the  value  of  h^       x=;-- ^ Z-V i — +^ 


•€C®^i>" 


SECTION  XI. 


i;OLUTION  OF  PROBLEMS  WHICH    CONTAIN 
TWO  OR  MORE  UNKNOWN  QUANTITIES. 

JDEMONSTRATION  OF  THEOREMS. 

Art.  322.  JJLN  the  example?  which  have  been  given  of 
the  resolution  of  equations,  in  the  preceding  sections,  each 
problem  has  contained  only  one  unknown  quantity-  Or  if, 
m  some  instances,  ther^  have  been  two,  they  have  been  sa 


1^  ALGEBRA. 

related  to  each  other,  that  they  have  both  been  expressed 
by  means  of  the  same  letter.     (Art.  195.) 

Silt  cases  frequently  occur  4n  which  several  unknown 
quantities  are  introduced  into  the  same  calculation.  And  if 
the  problem  is  of  such  a  nature,  as  to  admit  of  a  determin- 
ate answer,  there  will  arise  from  the  conditions,  as  many 
equations  independent  of  each  other,  as  there  are  unknown 
quantities. 

Equations  are  said  to  be  independent^  when  they  express 
different  conditions;  and  dependent,  when  they  express  the 
same  conditions  under  different  forms.  The  former  are  not 
convertible  into  each  other.  But  the  latter  may  be  chan- 
ged from  one  form  to  the  other,  by  the  methods  of  reduc- 
tion which  have  been  considered.  Thus  h  —  :rr=y,  and  h=y-\-x, 
are  dependent  equations,  because  one  is  formed  from  the 
other  by  merely  transposing  oc, 

323.  In  solving  a  problem,  it  is  necessary  first  to  find 
the  value  of  one  of  the  unknown  quantities,  and  then  of  the 
others  in  succession.  To  do  this,  we  must  derive  from  the 
equations  which  are  given,  a  new  equation,  from  which  all 
the  unknown  quantities  except  one  shall  be  excluded. 

Suppose  the  following  equations  are  given. 

1.  cr-f?/  =  14 

2.  a:^y  =  2. 

M  y  be  transposed  in  each,  they  will  become 

1.  0^=14— y 

2.  a:=2  +  i/. 

Here  the  first*  member  of  each  of  the  equations  is  x,  and 
the  second  member  of  each  is  equal  to  x.  But  according 
to  axiom  1 1  th,  quantities  which  arc  respectively  equal  to 
any  other  quantity  are  equal  to  each  other ;  therefore, 

Here  we  have  a  new^  equation,  which  contains  only  the 
unknown  quantity  y.     Hence, 

324  Rule  I.  To  exterminate  .one  of  two  unknown  quan- 
tities, and  deduce  one  equation  from  two  ;    Find  the  value 

OP  ONE  OF  THE  UNKNOWN  QUANTITIES  IN  EACH  OF  THE  EQUA- 
TIONS, AND  FORM  A  NEW  EQUATION  BY  MAKING  ONE  OF  THESE 
VALUES  EQUAL  TO  THE  OTHER. 

That  quantity  which  is  the  least  involved  should  be  the  one 
which  is  chosea  to  be  externainated. 


EQUATIONS.  ^S5 

For  the  convenience  of  referring  to  different  parts  of  a 
solution,  the  several  steps  will,  in  future,  be  numbered. 
When  an  equation  is  formed  from  one  immediately  preceding^ 
it  will  be  unnecessary  to  specify  it.  In  other  cases,  the  num- 
ber of  the  equation  or  equations  froi^  which  a  new  one  is. 
derived  will  be  referred  to. 

Prob.  1.  To  find  two  numbers  such,  that 
Their  sum  shall  be  24  ;  and 
The  greater  shall  be  equal  to  five  times  the  less. 

Let  j:'= the  greater ;  And  ?/= the  less. 

1.  By  the  first  condition,  ^+ ?/  =  24 

2.  By  the  second,  ^  ^=5^ 

3.  Transp.  yin  the  1st  equation,     x^^^A—y 

4.  Making  the  2d  and  3d  equal,      5y=:^4—y 

5.  And  ^=4,  the  less  numbei*. 

Prob.  2.  To  find  one  of  two  quantities, 
Whose  sum  is  equal  to  h  ;  and 
The  difference  of  whose  squares  is  equal  to  d. 

Let  x=ihe  greater  quantity ;  And  ^=the  lesi. 

L  By  the  first  condition,  x-\-y=^h       ^ 

2.  By  the  second,  x^—y^=d   V 

3.  Transp.  y^  in  the  2d  equation^  x^  =d-\-y^ 

4.  Bj  evolution,  (Art.  297.)  ^=  Vd-{-y'^ 

5.  Transp.  y  in  the  1st  equation,  x=:h—y 

6.  Making  the  4th  and  5th  equal  -^ d'\-y^=h—y 

h^-d 

7.  Therefore  yz=.—^--, 

Prob.  3.  Qiy^n  ax-^-hy—h-^  h  —  ad 

And  x+y^d\      To  find  y.     Ans.  3,=-^^^. 

325.  The  rule  given  above  may  be  generally  applied,  for 
the  extermination  of  unknown  quantities.  But  there  are 
cases,  in  which  other  methods  will  be  found  more  expedi- 
tious. , 

Suppose  x^=hy       7 

Ajod     ax'\-hx=^y^  \ 


25G  ALGEBRA. 

As  in  the  first  of  these  equations  x  is  equal  to  Ay,  we  may, 
in  the  second  equation,  substitute  this  value  of  x  instead  of 
X  itself.     The  second  equation  will  then  be  converted  into 

The  equality  of  the  two  sides  is  not  affected  by  this  alter- 
ation, because  we  only  change  one  quantity  ^,  for  another 
which  is  equal  to  it.  By  this  means  we  obtain  an  equation 
which  contains  only  one  unknown  quantity.     Hence,, 

326,  Rule  II.  To  exterminate  an  unknown  quantity,  find 

THE  VALUE  OF  ONE  OF  THE  UNKNOWN  QUANTITIES,  IN  ONE  OF 

THE  EQUATIONS  5  and  then,  in  the  other  equation,  substi- 
tute THIS  VALUE,  FOR  THE  UNKNOWN  QUANTITY   ITSELF. 

Prob.  4.  A  privateer  in  chase  of  a  ship  20  miles  distant, 
sails  8  miles,  while  the  ship  sails  7.  How  far  must  the  pri- 
vateer sail,  before  she  overtakes  the  ship  ? 

It  is  evident  that  the  whole  distance  which  the  privateer 
sails  during  the  chase,  must  be  to  the  distance  wbi<;h  the  ship 
sails  in  the  same  time,  as  8  to  7, 

Let  .v=the  distance  which  the  privateer  sails ; 
And  y=the  distance  which  the  ship  sails. 

1.  By  the  supposition,  a:=^+20) 

2.  And  also,  a: :  ^  ; :  8  : 7  V 

3.  Art.  188,  y=ix 

4.  Substituting  ''f  for  y  in  the  1  st  equation,  a;  =  |:c  +  20 

5.  Therefore  -       ji7=160. 

Prob.  5.  The  ages  of  two  persons  A  and  B  are  such,  that 
seven  years  ago,  A  was  three  times  as  old  as  B  ;  and  seven 
years  hence,  A  will  be  twice  as  old  as  B.  What  is  the  aee 
oiB?  ■ 

Let  ^=the  age  of  A  ;        And  y=i\\e  age  of  ^  ^ 
Then  x—1  was  the  age  of  A^  7  years  ago  ; 
And   y  —  1  was  the  age  of  B,  7  years  ago. 
Also  :r  -f.  7  will  be  the  age  of  A,  7  years  hence  ; 
And  y-\-l  will  be  the  age  of  B,  7  years  hence. 

1 .  By  the  first  condition,  at  —  7 =3  X  (^ — 7) = 3i/  —  2 1  > 

2.  By  the  second,  a;  +  7=2x(t^+7)=2y4-14  > 

3.  Transp.  7in  the  1st  equa.     x—3y  —  l^ 

4.  Subst.  3t/-.  1 4  foicTjin  the  2d,  3i^~I4+7=2y+14 
J.  Therefore  ^=21,  the  age  of  B» 


EQUATIONS.  157 

Frob.  6.  There  are  two  numbers^  of  whieb 

The  greater  is  to  the  less,  as  3  to  2  ;  and 
Their  sum  is  the  sixth  part  of  their  product. 
What  is  the  less  number  ?  Ans.  10. 

327.  There  is  a  third  method  of  exterminating  an  un- 
known quantity  from  an  equation,  which,  in  many  cases,  is 
preferable  to  either  of  the  preceding. 

Suppose  that  cc+3y=«? 
And  that         x  —  3i/—b  > 

If  we  add  together  the  first  members  of  these  two  equa- 
tions, and  also  the  second  members,  we  shall  have 

an  equation  which  contains  only  the  unknown  quantity  x. 
The  other,  having  equal  co-efficients  with  contrary  signs, 
has  disappeared.  (Art.  77.)  The  equality  of  the  sides  is 
preserved,  because  we  have  only  added  equal  quantities  to 
equal  quantities. 

Again,  suppose  3a?-h^=A   ) 
And  2x-{-y^=d   $ 

If  we  subtract  the  last  equation  from  the  first,  we  shall  have 

where  y  is  exterminated,  without  affecting  the  equality  of 
the  sides. 

Again,  suppose    x—2y=a'i 

And  jr-f4y=&5 

Multiplying  the  1st  by  2,  2^^4i/=2a 

Then  adding  the  2d  and  3d,     3sc=h+2a.     Hence, 

328.  Rule  III.  To  exterminate  an  unknown  quantity, 
Multiply  or  divide  the  equations,  if  necessary,  in 

such  a  manner  that  the  term  which  contains  one  op  the 
unknown  quantities  shall  be  the  same  in  both. 

Then  subtract  one  equation  from  the  other,  if  the 

SIGNS  OF  this  unknown  QUANTITY  ARE  ALIKE,  OR  ADD  THEM 
together,  IF  THE  SIGNS  ARE   UNLIKE. 

It  must  be  kept  in  mind  that  both  members  of  an  equa- 
tion are  always  to  be  increased  or  diminished,  multipHed  or 
4Jivided  alike.     (Art.  170.) 


I. 


158  ALGEBRA. 

Prob.  7.  The  numbers  in  two  opposing  armies  are  audi, 
that, 

The  sum  of  both  is  21 1 10  ;  and 

Twice  the  number  in  the  greater  army,  added  to  three 
times  the  number  in  the  less,  is  52219. 

What  is  the  number  in  the  greater  army  ? 

Let  a:=the  greater.  And  i^=the  less. 

1.  By  the  first  condition,  ^+y=21110 

2.  By  the  second,  2a;  +  3?/=52219 

3.  Multiplying  the  1st  by  3,  3a;+%=  63330 

4.  Subtracting  the  2d  from  the  3d,      a:=lllll. 

Prob.  8.  Given  2a? +^=16,  and  3x  — 3y=6,  to  find  the 
value  of  .V. 

1 .  By  supposition,  2a?+?/=16) 

2.  And  3x—3i/=-6  i 

3.  Multiplying  the  1st  by  3,  6a?-f33/=48 

4.  Adding  the  2d  and  3d,  dx=54: 

5.  Dividing  by  9,  x=6. 

Prob.  9,  Given  .t4-3/  =  14,  and  a,— ^=2,  to  find  the  value 
of?/-  Ans.  6. 

In  the  succeding  problems,  either  of  the  three  rules  for 
(exterminating  unknown  quantities  will  be  made  use  of,  as 
will  in  each  case  be  most  convenient. 

329.  When  one  of  the  unknown  quantities  is  determined, 
the  other  may  be  easily  obtained,  by  going  back  to  an  equa- 
tion which  contains  both,  and  substituting,  instead  of  that 
which  is  already  found,  its  numerical  value. 

Prob.  10.  The  mast  of  a  ship  consists  of  two  parts  : 
One  third  of  the  lower  part,  added  to  one  sixth  of  the 
upper  part,  is  equal  to  28  ;  and 

Five  times  the  lower  part,  diminished  by  six  times  the  up- 
per part,  is  equal  to  12, 

What  is  the  height  of  the  mast  ? 


EQUATIONS.  159. 

Let  a;=the  lower  part ;         And  y=the  upper  part. 

1.  By  the  first  condition,  ia;-|'  ^y=28  ) 

2.  By  the  second,  5.r--6y=12> 

3.  Multiplying  the  1st  by  6,  2^-\-i/=16B 

4.  Dividing  the  2d  by  6,  fa;  —y=2 

5.  Adding  the  3d  and  4th,  2Lr+|:c=17a 

6.  Multiplying  by  6,  12a?  +  5a7=1020 

7.  Uniting  terms,  and  dividing  by  17,  a; =60,  the  lower  part. 

Then  by  the  3d  step,  Six+y=16S 

That  is,  substituting  60  for  :r,  120+2/=l  68         [per  part. 

Transposing  120,  ?/=  168  — 120=48,  the  up- 

Prob.  1 1 .  To  find  a  fraction  such  tha^, 

If  a  unit  be  added  to  the  numerator,  the  fraction  will  be 
equal  to  i  ;  but 

If  a  unit  be  added  to  the  denominator,  the  fraction  will 
be  equal  to  i. 

Let  or = the  numerator,            And  y=the  denominator* 
1 .  By  the  first  condition,  =t  f 


4-1      *^ 


2.  By  the  second^  -77 

3.  Therefore  ;r=4,  the  numerator, 

4.  And  y=  15,  the  denominator. 

Prob.  1 2.  What  two  numbers  are  those, 

Whose  difference  is  to  their  sum,  as  3  to  3 ;  ai\d 
Whose  sum  is  to  their  product,  as  3  to  5  ? 

Ans.  10  and  2. 


Prob.  13.  To  find  two  numbers  such,  that 

The  product  of  their  sum  and  difference  shall  be  5,  and 
The  product  of  the  sum  of  their  squares  and  the  differ- 
ence of  their  squares  shall  be  65. 

Let  x—XhOi  greater  number  ;  And  y  the  less. 


160  ALGEBRA. 

1-  By  the  first  condition,  (•^+^)x(^'— Jr)=5           ^ 

2.  By  the  second,  (^H^^)  X  {x^  ~t/2)  =  65  > 

3.  Mult,  the  factors  in  the  1st,  (Art.  235,)     ^^  — «/^=5 

4.  Dividing  the  2d  by  the  3d,  (Art.  1 1 8.)     x^  +3^2^13 

5.  Adding  the  3d  and  4th,  2:^2  =18 

6.  Therefore  jt  =  3,  the  greater  number. 

7.  And  y=  2,  the  less. 

In  the  4th  step,  the  first  member  of  the  2d  equation  is  di- 
vided by  x^—y^^  and  the  second  member  by  5,  which  is 
equal  to  x^  ^y^. 

Prob.  14.  To  find  two  numbers,  whose  difference  is  8, 
and  product  240. 

Prob.  15.  To  find  two  numbers, 

Whose  difference  shall  be  12,  and 
The  sum  of  their  squares  1424.    ^ 

Let  a:  =  the  greater ;      '         And  ^=the  less. 

1.  By  the  1st  condition,  x—y^=\2 

2.  By  the  second,  x'^+y- 

3.  Transp.  y  in  the  1st,  a:=?/4-12 

4.  Squaring  both  sides,  a;^==y^  4-24^+144 

5.  Transp.  y^  in  the  2d,  a^*=1424— t/^ 

6.  Making  the  4th  and  5th  equal,    y^  +24y+ 144=1424— ya 

7.  Therefore,  ?/=— 6±  V676=— 6±26 
^^.  And  ;<'=3/4- 12=20+ 12=32. 

EQUATIONS  WHICH  CONTAIN  THREE  OR  MORE  UN- 
KNOWN QUANTITIES. 

330.  In  the  examples  hitherto  given,  each  has  contained 
no  more  than  tioo  unknown  quantities.  And  two  indepen- 
dent equations  have  been  sufficient  to  express  the  conditions 
of  the  question.  But  problems  may  involve  three  or  more 
unknown  quantities ;  and  may  require  for  their  solution  as 
many  independent  eqiititions. 

Suppose  oc+y-\-zz=il2     ^ 

And        x^2y— 2z=  1 0  >  are  given,  to  find  x,  y,  and  z. 

And         a^-|.y--2=4       3 


r3=:1424  5 


EQUATIONS.  161 

From  IJiese  three  equations,  two  others  may  be  derived, 
which  shall  contain  only  two  unknown  quantities.  One  of 
the  three  in  the  original  equations  may  be  exterminated,  in 
the  same  manner  as  when  there  are,  at  first,  only  two,  by 
the  rules  in  arts.  324,  6,  8. 

In  the  equations  given  above,  if  we  transpose  y  and  r,  we 
shall  have. 

In  the  first,        x=12— i/— j 

In  the  second,  x=l0—2i/+2z 

In  the  third,      x=4— y-f-z 

From  these  we  may  deduce  two  new  equations,  from 
which  X  shall  be  excluded. 


} 


By  making  the  1st  and  2d  equal,  12  — y— 2f  =  10--2y+22 

By  making  the  2d  and  3d  equal,  10 ^2y-\-2z =4— y  +  z 
R,educing  the  1st  of  these  two.  y==3z—2    > 

Reducing  the  second,  y—Z'\-6      ) 

From  these  two  equations,  one  may  be  derived  containing 
only  one  unknown  quantity, 

Making  one  equal  to  the  other,  3^—2=2"+ 6 

And  2^=4.     Hence, 

331.  To  solve  a  problem  containing  three  unknown  quanr 
tities,  and  producing  three  independent  equations. 

First,  from  the  three  equations  deduce  two,  con- 
taining ONLY  two  unknown  QUANTITIES, 

Then,  from  these  two  deduce  one,  containing  only  one 
unknown  quantity. 

For  making  these  reductions,  the  rules  already  given  are 
sufficient.     (Art.  324,  6,  8.) 

Prob.  16.  Let  there  be  given, 

1.  The  equation  x+5i/+6z  =  53i 

2.  And  ^  +  3y+ 3z=30  >  To  find  :r,  y,  and  z. 

3.  And  x-\'y  +  z  =  l2     3 

From  these  three  equations  to  derive  two,  containing  onh 
two  unknown  quantities, 

4.  Subtract  the  2d  from  the  1st,      2y4-3z=23  } 

5.  Subtract  the  3d  from  the  2d,      2j^+2r=:18  y 

From  these  two,  to  derive  one, 

6.  Subtract  the  5tk  from  the  4th,  r=5, 

'12 


J  (32  ALGEBRA. 

To  find  X  and  y,  we  have  only  to  take  their  values  from 
the  3d  and  5th  equations.     (Art.  329.) 

7.  Reducing  the  5th,  y=9-z=9-  5=4 

8.  Transposing  in  the  3d,  .:v=12--2:— 2/=12  — 5— 4=3. 

Prob.  17.  To  find  x^  ?/,  and  z,  from 

1.  The  equation  X'^y-\-z-=Vl     \ 

2.  And  x-f  2?^+3z=20( 

3.  And  Ja;-|-T^  +  ^=6    5 

4.  Multiplying  the  1st  by  3,  3x+3y-\-3z—3(i 

5.  Subtracting  the  2d  from  the  4th,    2:c-f  i/  =  16 

6.  Subtracting  the  3d  from  the  1st,    '^— -g-^-h^— 1^=^ 

7.  Clearing  the  6th  of  fractions,         4A;+3y=36  ") 

8.  Multiplying  the  5th  by  3,  6:r  + 3^=48  > 

9.  Subtracting  the  7th  from  the  8th,  2x_12.  Anda;=6. 

36 -4^'     36-24 

10.  Reducing  the  7th,  y=. — - — ■= — - — =4. 

11.  Reducing  the  1st,  2f==  12 -a: -y  =  12-6-4=2. 

In  this  example  all  the  reductions  have  been  made  accor- 
ding to  the  third  rule  for  exterminating  unknown  quantities. 
(Art.  328.)  But  either  of  the  three  may  be  used  at  pleas- 
ure. 

332.  A  calculation  may  often  be  very  much  abridged,  by 
the  exercise  of  judgment,  in  stating  the  question,  in  selec- 
ting the  equations  from  which  others  are  to  be  deduced,  in 
simplifying  fractional  expressions,  in  avoiding  radical  quanti- 
ties, &Cc  The  siiill  which  is  necessary  for  this  purpose,  how- 
ever, is  to  be  acquired,  not  from  a  system  of  rules;  but 
from  practice,  and  a  habit  of  attention  to  the  peculiarities  in 
the  conditions  of  different  problems,  the  variety  of  ways  in 
which  the  same  quantity  may  be  expressed,  the  numerous 
forms  which  equations  may  assume,  &;c.  In  many  of  the 
examples  in  this  and  the  preceding  sections,  the  processes 
might  have  been  shortened.  But  the  object  has  been  to  il- 
lustrate general  principles,  rather  than  to  furnish  specimens 
of  expeditious  solutions.  The  learner  will  do  well,  as  he 
passes  along,  to  exercise  his  skill  in  abridging  the  calcula- 
tions wliich  are  here  given,  or  substituting  others  in  their 
stead. 


EQUATIONS.  k;:*^ 

Prob.  18.  Given,  <2.  .t+2:=£6>    To  find  x^  ^,  and  z, 
(3.  ij-{-z=.c>^ 

a'{-h—c                 a  +  c—b                    b+c—a 
Ans.  :t== ^ — .  And  y= ^ — .     And  2'= — ^ 

Prob.  19.  Three  persons,  A,  B,  and  C,  purchase  a  horse 
for  100  dollars,  but  neither  is  able  to  pay  for  the  whole. 
The  payment  would  require. 

The  whole  of  .^'s  money,  together  with  half  of  B's  ;  or 

The  whole  of  Bh,  with  one  third  of  C's  *,  or 

The  whole  of  C's,  with  one  fourth  of  Ah, 
How  much  money  had  each  ? 

Let:r=^'s  z=:Os 

y==Bh  a=100 

By  the  first  condition,  sc-\-~y=a^ 

By  the  second,  y-\-^z=:a> 

By  the  third,  z-\-^x=a^ 

Therefore       .r=64.  ^=72.           z=S4. 

333.  The  learner  must  exercise  kis  own  judgment,  as  to 
the  choice  of  the  quantity  to  be  first  exterminated.  It  will 
generally  be  best  to  begin  with  that  which  is  most  free  from 
co-efficients,  fractions,  radical  signs,  &:c. 

Prob.  20.  The  sum  of  the  distances  which  three  persons, 
A,  B,  and  C,  have  travelled  is  62  miles  ; 
Ah  distance  is  equal  to  4  times  C's,  added  to  twice  Bh  ;  and 
Twice  .^'s  added  to  3  times  Bh,  is  equal  to  1 7  times  C's. 

What  are  the  respective  distances  ? 

Ans.  ^'s,  46  miles ;  Bh,  9  ;  and  C's,  7. 

Prob.  21.  To  find  x,  y,  and^,  from 

The  equation  J^+-|y-f  J2^=62) 

And  ix+ii/+iz=:47\ 

And  ix+iy+iz=3S} 

And  ;v=24.         y— 60.         z=120. 

fcCxy=600^ 
Prob.  22.  Given  ^xz=dOO\    To  find  x,  ij,  and  z. 
(yz  =200) 
Ans.  a:=30.  ^      y=20.  5^=10. 

k 


l;g4  ALGEBRA. 

334.  The  same  method  which  is  employed  for  the  reduc- 
tion of  three  equations,  may  be  extended  to  4,  5,  or  any 
number  of  equations,  containing  as  many  unknown  quanti- 
ties. The  unknown  quantities  may  be  exterminated,  one 
after  another,  and  the  number  of  equations  may  be  reduced 
by  successive  steps,  from  five  to  four,  from  four  to  three, 
from  three  to  two,  &:c. 

Prob.  23.  To  find  a?,  ^,  y,  and  z,  from 
1.  The  equation  |^y+2;+|zo  =  8^ 

i.  And  xtjS=?2    [^""'•equations. 

4.  And  a:-{-w+Zi=10  J 

5.  Clear,  the  1st  of  frac.    y+2z+w  =  ^6y 

6.  Subtract.  2d  from  3d,  z^w='3    >  7%ree  equations. 

7.  Subtract.  41h  from  3d,  y—w  =  2    ) 

8.  Adding  5th  and  6th,  y+3z  =  19}    rp              r 

9.  Subtract.  7th  from  6th,  ~2/+^  =  l  5   ^^'^  ^^^^^^^^^' 

10.  Adding  8th  and  9th,  4z=z20      Orz  =  5  J 

11.  Transp.  in  the  8th,  y=ld—Sz  =  4:         '    Quantities 

12.  Transp.  in  the  3d,  jo^l'2—y—z=3     (      required. 

13.  Transp.  in  the  2d,  w  =  9—x—y=2      ) 

Prob.  24.  Given   ^'^^+!f^=^2/  L  To  find  w,  x,  y,  and 

;^+195  =  3a; 
Answer.  5^=100  y=:90 

^  =  150  z=l05. 

Prob.  25.  There  is  a  certain  number  consisting  of  two 
digits.  The  left-hand  digit  is  equal  to  3  times  the  right-hand 
digit ;  and  if  twelve  be  subtracted  from  the  number  itself, 
the  remainder  will  be  equal  to  the  square  of  the  left-hand 
digit.     What  is  the  number  ? 

Let  A;=the  left-hand  digit,  and  y=the  right-hand  digit. 

As  the  local  value  of  figures  increases  in  a  tenfold  ratio 
from  right  to  left ;  the  number  required  =\Ox-\-y 

By  the  conditions  of  the  problem  x  =  3y} 

And  10a:+^-12=;^2^ 

The  required  number  is,  therefore,  93. 


EQUATIONS.  ^.-  165     '^Tn?^ 

Prob.  26.   If  a  certain  numbei  be  divided  by  we  product     '^^ 
of  its  two  digits,  the  quotient  will  be  2  ;  and  if  Ti  be  ad-     i^.SI'! 
ded  to  the  number,  the  digits  will  be  inverted.     What  is  tli^^, 
number  ?  Aiis.  36, 

Prob.  27.  There  are  two  numbers,  such,  that  if  itfie  less 
be  taken  from  3  times  the  greater,  the  remaindei  will  be 
S5 ;  and  if  4  times  the  greater  be  divided  by  3  times  the 
less+1,  the  quotient  will  be  equal  to  the  less.  What  are 
tiie  numbers  ?  Ans.  13  and  4. 

Prob.  28.  There  is  a  certain  fraction,  such,  that  if  3  be 
added  to  the  numerator,  the  value  of  the  fraction  will  be  ^  ; 
but  if  1  be  subtracted  from  the  denominator,  the  value  will 
be  ^.     What  is  the  fraction  ?  A         "^ 

"^'  21* 

Prob.  29.  A  gentleman  has  two  horses,  and  a  saddle 
which  is  worth  10  guineas.  If  the  saddle  be  put  on  the 
first  horse,  the  value  of  both  will  be  double  that  of  the 
second  horse  ;  but  if  the  saddle  be  put  on  the  second  horse, 
the  value  of  both  will  be  less  than  that  of  the  first  horse 
by  13  guineas.     What  is  the  value  of  each  horse  ? 

Ans.  56  and  33  guineas. 

Prob.  30.  Divide  the  number  90  into  4  such  parts,  that 
the  first  increased  by  2,  the  second  diminished  by  2,  .the 
third  midtiplcd  by  2,  and  the  fourth  divided  by  2,  shall  all  be 
equal. 

If  x^  ?/,  and  z,  be  three  of  the  parts,  the  fourth  will  be 
^O—x—y—z,     And  by  the  conditions, 

x4-2=y— 2 
x  +  2  =  22r 

90— jc— «— 2; 

The' parts  required  are  18,  22,  10,  and  40. 

Prob.  31.  Find  three  numbers,  such,  that  the  first  with  i 
the  sum  of  the  second  and  third  shall  be  1 20 ;  the  second 
with  I  the  difference  of  the  third  and  first  shall  be  70  j  and 
-}  the  sum  of  the  three  numbers  shall  be  95. 

Prob.  32.  What  two  numbers  are  those,  whose  difference, 
«um,  and  product,  are  as  the  numbers  2,  3,  and  5  ? 

Ans.   10  and  2. 


IQ^  ALGEBRA. 

Prob.  33.  A  Vintner  sold,  at  one  time,  20  dozen  of  port 
wine,  and  30  dozen  of  sherry  ;  and  for  the  whole  received 
120  guineas.  At  another  time,  he  sold  30  dozen  of  port, 
and  25  dozen  of  sherry,  at  the  same  prices  as  before  ;  and  for 
the  whole  received  140  guineas.  What  was  the  price  of  a 
dozen  of  each  sort  of  wine  ? 

Ans.  The  port  was  3  guineas,  and  the  sherry  2  guineas  a 
dozen. 

Prob.  34.  A  merchant  having  mixed  a  certain  number 
©f  gallons  of  brandy  and  water,  found  that,  if  he  had  mixed 
6  gallons  more  of  each,  he  would  have  put  into  the  mix- 
ture 7  gallons  of  brandy  for  every  6  of  water.  But  if  he 
had  mixed  6  less  of  each,  he  would  have  put  in  6  gallons  of 
brandy  for  every  5  of  water.  How  many  of  each  did  he 
mix  ? 

Ans.  78  gallons  of  brandy,  and  66  of  water. 

Prob.  S5.  What  fraction  is  that,  whose  numerator  being 
doubled,  and  the  denominator  increased  by  7,  the  value  be- 
comes f  ;  but  the  denominator  being  doubled,  and  the  nu- 
merator increased  by  2,  the  value  becomes  f  ?        Ans.  -J. 

*  Prob.  36.  A  person  expends  30  cents  in  apples  and  pears^ 
giving  a  cent  for  four  apples,  and  a  cent  for  5  pears.  He 
afterwards  parts  with  half  his  apples  and  one  third  of  his 
pears,  the  cost  of  which  was  13  cents.  How  many  did  he 
buy  of  each  ?  Ans.  72  apples,  and  60  pears. 


335.  If  in  the  algebraic  statement  of  the  conditions  of  a 
problem,  the  original  equations  are  more  numerous  than  the 
unknown  quantities  ;  these  equations  will  either  be  contra- 
dictory.  or  one  or  more  of  them  will  be  stiperfluous. 

Thus  the  equations  <  i^Zloo  (  ^^^  contradictory. 

For  by  the  first,  03=20,  while  by  the  second,  a; =40. 

But  if  the  latter  be  altered,  so  as  to  give  to  x  the  same 
value  as  the  former,  it  will  be  useless,  in  the  statement  of  a 

*  For  more  examples  of  the  solution  of  Problems  by  equations,  see  Eu- 
ler's  Algebra,  Part.  I.  Sec.  4,  Simpson's  Algebra,  Sec.  II,  Simpson's  Exer- 
cises, Maclaurin's  Algebra,  Part  I,  Chap.  2  and  13,  Emerson's  Algebra, 
Book  II,  Sec.  1,  Saunderson's  Algebra,  Book  II  and  III,  Dodson's  Mathe- 
matical Repository,  and  Bland's  Algebraical  Problems. 


EQUATIONS.  J  07 

problem.     For  nothing  can  be  determined  from  the  one, 
which  can  not  be  from  the  other. 

Thus,  of  the  equations    j  i^Zin  t  ^^^  ^^  superfluous. 

For  either  of  them  is  sufficient  to  determine  the  value  of 
X,  They  are  not  independent  equations.  (Art.  322.)  One  i« 
convertible  into  the  other.  For  if  we  divide  the  1st  by  6, 
it  will  become  the  same  as  the  second. 

Or  if  we  multiply  the  second  by  6,  it  will  become  tha 
same  as  the  first. 

336.  But  if  the  number  of  independent  equations  produ- 
ced from  the  conditions  of  a  problem,  is  less  than  the  num- 
ber of  unknown  quantities,  the  subject  is  not  sufficiently 
limited  to  admit  of  a  definite  answer.  For  each  equation 
can  limit  but  one  quantity.  And  to  enable  us  to  find  this 
quantity,  all  the  others  connected  with  it,  must  either  be 
previously  known,  or  be  determined  from  other  equations. 
If  this  is  not  the  case,  there  will  be  a  variety  of  answers 
which  will  equally  satisfy  the  conditions  of  the  question.  If, 
for  instance,  in  the  equation 

a:-fy  =  100, 
so  and  y  are  required,  there  may  be  fifty  different  answers. 
The  values  of  x  and  y  may  be  either  99  and  1,  or  98  and  2, 
or  97  and  3,  &;c.  For  the  sum  of  each  of  these  pairs  of 
numbers  is  equal  to  100.  But  if  there  is  a  second  equation 
which  determines  one  of  these  quantities,  the  other  may 
then  be  found  from  the  equation  already  given.  As  x+y  =  100, 
if  vT  =  46,  y  must  be  such  a  number  as  added  to  46  will  make 
100,  that  is,  it  must  be  54.  No  other  number  will  answer 
this  condition. 

337.  For  the  sake  of  abridging  the  solution  of  a  problem, 
however,  the  number  of  independent  equations  actually  put 
upon  paper  is  frequently  less,  than  the  number  of  unknown 
quantities.  Suppose  we  are  required  to  divide  100  into  two 
such  parts,  that  the  greater  shall  be  equal  to  three  times  the 
less.  If  we  put  x  for  the  greater,  the  less  will  be  100~^. 
(Art.  195.) 

Then  by  the  supposition,  a'  =  300— 3a;. 

Transposing  and  dividing,        x  =  '75,  the  greater. 
And  1 00  —  75  =  25,  the  less. 


168  ALGEBRA. 

Here,  two  unknown  quantities  are  found,  although  therer 
appears  to  be  but  one  independent  equation.  The  reason  of 
this  is,  that  a  part  of  the  solution  has  been  omitted,  because 
it  is  so  simple,  as  to  be  easily  supplied  by  the  mind.  To  have 
a  view  of  the  whole,  without  abridging,  let  x  =  the  greater 
number,  and  y  =  the  less. 

1.  Then  by  the  supposition,  a;+y=100> 

2.  And  3y=x  3 

3.  Transposing  x  in  the  1st,  y  =  100—x 

4.  Dividing  the  2d  by  3,  ^=7^ 

5.  Making  the  3d  and  4th  equal,  ^a;  =  1 00 — a; 

6.  Multiplying  by  3,  a;  =  300  — 3a; 

7.  Transposing  and  dividing,  x  =  75,  the  greater. 

8.  By  the  3d  step,  y=:  100— a? =25,  the  less* 

By  comparing  these  two  solutions  with  each  other,  it  will 
be  seen  that  the  first  begins  at  the  6th  step  of  the  latter,  alt 
the  preceding  parts  being  omitted,  because  they  are  too  sim- 
ple to  require  the  formality  of  writing  down. 

Prob.  To  find  two  numbers  whose  sum  is  30,  and  the  dif- 
ference of  their  squares  1 20. 

Let«=30  5  =  120 

a; = the  less  number  required.  . 
Then  «— a;=the  greater.     (Art.  l95.) 

And  a^  —2ax-\-x^  =the  square  of  the  greater.  (Art.  214.) 
From  this  subtract  x^  the  square  of  the  less,  and  we  shalt 
have  a'— 2aa;=the  difference  of  their  squares. 

a^-b     (30)* -120 
Therefore     x=-^^=      2x30~=^^- 

338.  In  most  cases  also,  the  solution  of  a  problem  which 
contains  many  unknown  quantities,  may  be  abridged,  by  par- 
ticular artifices  in  substituting  a  single  letter  jfor  several. 
(Art.  321.) 

"*  Suppose  four  numbers,  u,  x,  ?/,  and  z,  are  required,  of  which 
The  sum  of  the  three  first  is  13 

The  sum  of  the  two  first  and  last  17 

The  sum  of  the  first  and  two  last  18 

The  sum  of  the  three  last  21 

*  LudlamV  Als^ebra,  art.  161. c. 


EQUATIONS.  169 

Then    1.  w+x+y=13 

2.  u+x+z— 17 

3.  u+y+z==lS 

I^et  S  be  substituted  for  the  sum  of  the  four  numbers,  that 
is,  for  u+X'\~y-\-z,     It  will  be  seen  that,  of  these  four  equa- 
tions, 
The  first  contains  all  the  letters  except  z,  that  is,  S—z=l  3 
The  second  contains  all  except  y,  that  is,  S— y=17 

The  third  contains  all  except  x,  that  is,  S—x=:'iS 

The  fourth  contains  all  except  u,  that  is,  S— m=21. 

Adding  all  these  equations  together,  we  have 
4^8— z—y—x— 11=69 
Or     45--(5;+2/+a;+w)=69  (Art.  88.c.) 
But     S=(z-\-y-{-x-\-u)  by  substitution. 
Therefore,   4S-S=69,  that  is,  35=69,  and  5=23. 

Then  putting  23  for  5,  in  the  four  equations  in  which  it  is 
first  introduced,  we  have 

23-z=13")  rz=23- 13  =  10 

23-2,  =  17  f  ^^,^^^,^  >  2/=23-17=6 
23-a:=18  C  ^^^^^^^^^  ^  a;  =  23-18  =  5 
23-M  =  21J  Cw=23-21=2. 

Contrivances  of  this  sort  for  facilitating  the  solution  of 
particular  problems,  must  be  left  to  be  furnished  for  the  oc- 
casion, by  the  ingenuit}'  of  the  learner.  They  are  of  a  na- 
ture not  to  be  taught  by  a  system  of  rules. 

339.  In  the  resolution  of  equations  containing  several  un* 
known  quantities,  there  will  often  be  an  advantage  in  adopt- 
ing the  following  method  of  notation. 

The  co-efficients  of  one  of  the  unknown  quantities  are 
represented, 

In  the  Jirst  equation,  by  a  single  letter,  as  a. 
In  the  second,  by  the  same  letter  marked  with  an  accent,  as  a', 
In  the  third,  by  the  same  letter  with  a  double  accent,  as  a", 

&c. 

The  co-efficients  of  the  other  unknown  quantities,  are 
jFcpresented  by  other  letters  marked  in  a  similar  manner ; 
&8  are  also  the  termjB  which  consist  of  known  quantities 
only. 

23 


170  ALGEBRA. 

Two  equations  containing  the  two  unknown  quantities  os- 
and  y  may  be  written  thus, 

aX'{-hy:rzc 

a'X'\-h'y—c' 

Three  equations  containing  a?,  y,  and  z,  thus^ 
ax-\-by-{-cz=d 
a'x-^-b'y-^-c'z^d' 
a"x+l"y-\-c"z=d" 

Four  equations  containing  a:,  y,  z,  and  w,  thus 
ax-^-  by'{-cz-^  dii = c 
a'x + b'y  -f  c'-sr + 6?'w=  e' 

The  same  letter  is  made  the  co-efficient  of  the  same  un- 
known quantity,  in  different  equations,  that  the  co -efficients 
of  the  several  unknown  quantities  may  be  distinguished,  in 
any  part  of  the  calculation.  But  the  letter  is  marked  with, 
different  accents,  because  it  actually  stands  for  different  quan- 
tities. 

Thus  we  may  put  «=4,     a'=:6,     a"=zlOy     a'"=20,  ire. 

To  hnd  the  value  of  x  and  y. 

1.  In  the  equation,  ax-\-by=c    > 

2.  And  a'x-\-b'y=:c' ) 

3.  Multiplying  the  1st  by  6',  (Art.  328.)  ab'x-^bb'y=cb' 
A,  Multiplying  the  2d  by  6,  ba'x-\-bb'y=ibc' 

5.  Subtracting  the  4th  from  the  3d,       ab'x—ba'x=.cb'—bc' 

cb'—bc''\ 

6.  Dividingby  «&'  —  &«'  (Art.  121.)       x—    .,_,   ,  j 

ac' —ca'^ 
By  a  similar  process,  y=^j73^j 

The  symmetry  of  these  expressions  is  well  calculated  to 
fix  them  in  the  memory.  The  denominators  are  the  same 
in  both ;  and  the  numerators  are  like  the  denominators,  ex- 
cept a  change  of  one  of  the  letters  in  each  term.  But  the 
particular  advantage  of  this  method  is,  that  the  expressions 
here  obtained  may  be  considered  as  general  solutions,  which 
give  the  values  of  the  unknown  quantities,  i»  other  equa- 
tions, of  a  similar  nature. 


EQUATIONS.  1 71 

Thus  if  10x  +  62/=:100> 
And         40a?+4y=200  5 
Then  putting  a  =  10  5=6  c=100 

a'=:40  i'  =  4  c'=;200 

c6'-6c'       100x4-6x200 

We  have  a; = -77 — r~^  =   ,^,,  . — ^TTatT = 4 . 
ao—oa        10x4  —  6x40 

ac'  —  ca'      10x200-100x40 

And  2/=;;57Z6;^---T0^4~6^0~  =  ^^- 

The  equations  to  be  resolved  may,  originally,  consist  of 
more  than  three  terms.  But  if  they  are  of  the  first  degree, 
and  have  only  two  unknown  quantities,  each  may  be  redu- 
Ged  to  three  terms  by  substitution. 

Thus  the  equation  d^c—Ax+hy—Qy  =m  +  8 

Is  the  same,  by  art.  120,  as  ((7— 4)a?+(A— 6)y=m+8. 

And  putting    a=c?— 4,  b=h—6               c=m+8 

It  becomes  ax-\-by=c,* 

DEMONSTRATION  OF  THEOREMS. 

340.  Equations  have  been  applied,  in  this  and  the  prece- 
ding sections,  to  the  solution  of  problems.  They  may  be 
employed  with  equal  advantage,  in  the  demonstration  of  the- 
orems. The  principal  difference,  in  the  two  cases,  is  in  the 
order  in  which  the  steps  are  arranged.  The  operations 
themselves  are  substantially  the  same.  It  is  essential  to  a 
demonstration,  that  complete  certainty  be  carried  through 
every  part  of  the  process.  (Art.  11.)  This  is  effected,  in 
the  reduction  of  equations,  by  adhering  to  Uie  general  rule, 
to  make  no  alteration  which  shall  affect  the  value  of  one  of 
the  members,  without  equally  increasing  or  diminishing  the 
other.  In  applying  this  principle,  we  are  guided  by  the  ax- 
ioms laid  down  in  art.  63.  These  axioms  are  as  applicable 
to  the  demonstration  of  theorems,  as  to  the  solution  of  prob- 
lems. 

But  the  order  of  the  steps  will  generally  be  different.  In 
solving  a  problem,  the  object  is  to  find  the  value  of  the  un- 
known quantity,  by  disengaging  it  from  all  other  quantities. 
But  in  conducting  a  demonstration,  it  is  necessary  to  bring 

*  For  the  application  of  this  plan  of  notation  to  the  solution  of  equations 
which  contain  more  than  two  unknown  quantities,  see  La  Croix's  Algebra, 
art.  85,  Maclaurin's  Algebra,  Part  I.  Chap.  12,  Fenn's  Algebra,  p.  57,  and 
a  paper  of  Laplace,  in  the  Memoir*  oi  the  Acadenay  of  Sciences  for  1772, 


1 72  ALGEBRA. 

the  equation  to  that  particular  form  which  will  express,  in 
algebraic  terms,  the  proposition  to  be  proved. 

Ex.  1.  Thearem,  Four  times  the  product  of  any  two 
numbers,  is  equal  to  the  square  of  their  sum,  diminished  by 
the  square  of  their  difference. 

Let  a;=the  greater  number,  5= their  sum, 

y= the  less,  J = their  difference. 

Demonstration. 

1.  By  the  notation  x+y:=s   > 

2.  And  x—y=d  J 

3.  Adding  the  two,  (Ax.  I.)  2a;=5-f^ 

4.  Subtracting  the  2d  from  the  1st,  2y=s—d 

5.  Mult.  3d  and  4th,  (Ax.  3.)  4a:y={s-{-d)x{s—d) 

6.  That  is,  (Art.  235.)  4xy=s^-d'' 

The  last  equation  expressed  in  words  is  the  proposition 
which  was  to  be  demonstrated.  It  will  be  easily  seen  that 
it  is  equally  applicable  to  any  two  numbers  whatever. 
For  the  particular  values  of  x  and  y  will  make  no  difference 
in  the  nature  of  the  proof. 

Thus  4x8x6=(8  +  G)2-(8-6)  =  192. 
And    4xl0x6=(10+6)*-"(10-6)2=:240. 
And    4X  12X  10=(l2-(-10)2  -(12-10)^=480. 

Theorem  2.  The  sum  of  the  squares  of  any  two  numbers 
is  equal  to  the  square  of  their  difference,  added  to  twice 
their  product. 

Let  a:=the  greater,  e7==  their  difference. 

^=the  less^  />=their  product. 

Demonstration, 

1.  By  the  notation  ar— y=c?) 

2.  And  ^y=P      y 

3.  Squaring  the  first,  x^'-2xy+y^^d^ 

4.  Multiplying  the  2d  by  2,  2xy=2p 

5.  Adding  the  3d  and  4th,  x^  -^y^  =d^  +  2p, 

Thus  10*  +  8«=(10-8)*+2xl0x8  =  164. 

341.  General  propositions  are  also  discovered^  in  an  expe- 
ditious manner,  by  means  of  equations.  The  relations  of 
quantities  may  be  presented  to  our  view,  in  a  great  variety 


RATIO.  173 

i)f  ways,  hy  the  several  changes  through  which  a  given  equa- 
tion may  be  made  to  pass.  Each  step  in  the  procelfe  will 
contain  a  distinct  proposition. 

Let  s  and  d  be  the  sum  and  difference  of  two  quantities 
X  and  y,  as  before. 

1.  Then  s=xi-y} 

2.  And  d=x^y  > 

3.  Dividing  the  1st  by  2,  |^=i^'~i^ 

4.  Dividing  the  2d  by  2,  \d=\x—\y 

5.  Adding  the  3d  and  4th,  \s-\-\d~^X'\-^x=^x 

6.  Sub.  the  4th  from  the  3d,  y—\d=ly'\-\y=y. 

That  is, 

Half  the  difference  of  two  quantities,  added  to  half  their 
sum,  is  equal  to  the  greater  ;  and 

Half  their  difference  subtracted  from  half  their  sum^is  equal 
to  the  less* 


•«^®^^- 


SECTION  XII. 


RATIO  AND  PROPORTION.* 

Art.  342.  X  HE  design  of  mathematical  investigations, 
is  to  arrive  at  the  knowledge  of  particular  quantities,  by 
comparing  them  vrith  other  quantities,  either  equal  to,  or 
greater,  or  less  than  those  which  are  the  objects  of  inquiry. 
The  end  is  most  commonly  attained  by  means  of  a  series  of 
equations  and  proportions*     When  we  make  use  of  equa- 

*  Euclid's  Elements,  Book  5,  7,  8.  Euler's  Algebra,  Part  I,  Sec.  3. 
Emerson  on  Proportion.  Camus'  Geometry,  Book  III.  Ludlam's  Mathe- 
matics. Wallis'  Algebra,  Chap.  19,  20.  Saunderson's  Algebra,  Book  7. 
Barrow's  Mathematical  Lectures.     Analyst  for  March,  1814, 


k 


174  ALGEBRA. 

tions,  we  detemiine  the  quantity  sought,  hy  tliscoveHng  m 
e^uaWkj  with  some  other  quantity  or  quantities  already 
known. 

We  have  frequent  occasion,  however,  to  compare  the  un- 
known quantity  with  others  which  are  not  equal  to  it,  but  ei- 
ther greater  or  less.  Here,  a  difTerent  mode  of  proceeding 
t)ecomes  necessary.  We  may  inquire,  either  Jiow  much  one 
of  the  quantities  is  greater  than  the  other ;  or  how  many  times 
the  one  contains  the  other.  In  finding  the  answer  to  either 
of  these  inquiries,  we  discover  what  is  termed  a  ratio  of  the 
two  quantities.  One  is  called  arithmetical^  and  the  other 
geometrical  ratio.  It  should  be  observed,  however,  that  both 
<hese  terms  have  been  adopted  arbitrarily,  merely  for  dis- 
tinction sake.  Arithmetical  ratio,  and  geometrical  ratio,- 
are  bol'h  of  them  applicable  to  arithmetic,  and  both  to  geom- 
etr}^ 

As  the  whole  of  the  extensive  and  important  subject  of 
proportion  depends  upon  ratios,  it  is  necessary  that  these 
should  be  clearly  and  fully  understood. 

343.  Arithmetical  ratio  is  the  difference  between  two 
quantities  or  sets  of  quantities.  The  quantities  themselves 
are  called  the  terms  of  the  mtio,  that  is,  the  terms  between 
which  the  ratio  exists.  Thus  two  is  the  arithmetical  ratio 
of  5  to  3.  This  is  sometimes  expressed,  by  placing  two 
points  between  the  quantities  thus,  5  .  .  3,  which  is  the  same 
as  5—3.  Indeed  the  term  arithmetical  ratio,  and  its  nota- 
tion, by  points  are  almost  needless.  For  the  one  is  only  a 
substitute  for  the  word  difference,  and  the  other  for  the 
sign  — . 

341.  If  both  the  terras  of  an  arithmetical  ratio  be  multi- 
plied or  dhided  by  the  same  quantity,  the  ratio  will,  in  effect, 
be  multiplied  or  divided  by  that  quantity. 

Thus  if  a-b=r 

Then  mult,  both  sides  by  h,  (Ax.  3.)  ha—hh—hr 

a       b       r 
And  dividing  by  h,  (Ax.  4.)  "^  '~~h^^~k* 

345.  If  the  terms  of  one  arithmetical  ratio  be  added  to, 
or  subtracted  from,  the  corresponding  terms  of  another,  the 
ratio  of  their  sum  or  difference  will  be  equal  to  the  sum  or 
dlfft^reiice  of  the  two  ratios. 


RATIO.  J  75 

And  J-^S^^'^*^^^^^'^^^^^' 

Then  (a-\-d)-{b-{'k)={a-b)+{d-h).  For  each=:«4^-6-^. 

And  {a—d)-{h-h)^(a--h)  —  {d-h).  For  each=a-J-Zr-j-A, 
Thus  the  arith.  ratio  of  1 1 . .  4  is  7  > 
And    the  arith.  ratio  of    5  . .  2  is  3  ) 

The  ratio  of  the  sum  of  the  terms  16 .. 6  is  10,  the  sum  of 

the  ratios. 
The  ratio  of  the  difference  of  the  terms  6  . .  2  is  4,  the  dif» 

ference  of  the  ratios. 

346.  Geometrical  ratio  is  that  relation  between 
quantities  which  is  expressed  by  the  quotient  of  the 
one  divided  by  the  other.* 

Thus  the  ratio  of  8  to  4,  is  f  or  2.  For  this  is  the  quo- 
tient of  8  divided  by  4.  In  other  words,  it  shows  how  often 
4  is  contained  in  8. 

In  the  same  manner,  the  ratio  of  any  quantity  to  another 

may  be  expressed  by  dividing  the  former  by  the  latter,  or, 

which  is  the  same  thing,  maldng  the  former  the  numerator 

of  a  fraction,  and  the  latter  the  denominator. 

a 
Thus  the  ratio  of  a  to  5  is  t". 

d^h 

The  ratio  of  d-^-h  to  ^+t^)is-7T7. 

347.  Geometrical  ratio  is  also  expressed  by  placing  two 
points,  one  over  the  other,  between  the  quantities  compared. 

Thus  a :  b  expresses  the  ratio  of  a  to  ^  ;  and  12:4  the 
ratio  of  12  to  4.  The  two  quantities  together  are  called  a 
couplet^  of  which  the  first  term  is  the  antecedent,  and  the 
last,  the  consequent* 

348.  This  notation  by  poiftts,  and  the  other  in  the  form  of 
a  fraction,  may  be  exchanged  the  one  for  the  other,  as  con- 
venience may  require  ;  observing  to  make  the  antecedent  of 
the  couplet,  the  numerator  of  the  fraction,  and  the  conse- 
quent the  denominator. 

h 
Thus  10  : 5  is  the  same  as  V>  and  h  :  J,  the  same  as  -j. 

349.  Of  these  three,  the  antecedent,  the  consequent,  an^^, 
the  ratio,  any  two  being  given,  the  other  may  be  found, 

*  §^e  Note  H. 


176  ALGEBRA, 

Let  «=the  antecedent,  c=the  consequent,  r=the  ratio. 


a 


By  definition  '"=="7' ;  that  is,  the  ratio  is  equal  to  the  antece- 
dent divided  by  the  consequent. 

Multiplying  by  c,  a—cr^  that  is,  the  antecedent  is  equal  to 
the  consequent  multiplied  into  the  ratio. 

.  .  a 

Dividing  by  r,  c=— ,  that  is,  the  consequent  is  equal  to  the 

antecedent  divided  by  the  ratio. 

Cor.  1.  If  two  couplets  have  their  antecedents  equal,  and 
their  consequents  equal,  their  ratios  must  be  equal.* 

Cor.  2.  If,  in  two  couplets  the  ratios  are  equal,  and  the 
antecedents  equal,  the  consequents  are  equal  :  and  if  the 
ratios  are  equal  and  the  consequents  equal,  the  antecedents 
are  equal. t 

350.  If  the  two  quantities  compared  are  equal,  the  ratio 
is  a  unit,  or  a  ratio  of  equality.  Thus  the  ratio  of  3  X  6  : 1 8 
is  a  unit,  for  the  quotient  of  any  quantity  divided  by  itself 
is  1. 

If  the  antecedent  of  a  couplet  is  greater  than  the 
consequent,  the  ratio  is  greater  than  a  unit.  For  if  a  divi- 
dend is  greater  than  its  divisor,  the  quotient  is  greater  than 
a  unit.  Thus  the  ratio  of  18  :  6  is  a  (Art.  128.  cor.)  This 
is  called  a  ratio  of  greater  inequality* 

On  the  other  hand,  if  the  antecedent  is  less  than  the  con- 
sequent, the  ratio  is  less  than  a  unit,  and  is  called  a  ratio  of 
less  inequality.  Thus  the  ratio  of  2  :  3,  is  less  than  a  unit, 
because  the  dividend  is  less  than  the  divisor. 

351.  Inverse  or  reciprocal  ratio  is  the  ratio  op  the 
RECIPROCALS  OF  TWO  QUANTITIES.     See  art.  49. 

Thus  the  reciprocal  ratio  of  6  to  3,  is  \  to  -|,  that  is  ^-^|. 

a 
The  direct  ratio  of  a  to  6  is  "T",  that  is,  the  antecedent 

divided  by  the  consequent. 

rry^  .  ,  .       .      1      1  1  1  1  Z^  5 

I  he  reciprocal  ratio,  is  — :t  or— -r-r=~^  X-r=~5 
^  ^      a    b       a      b      a       \       a^ 

that  is,  the  consequent  h  divided  by  the  antecedent  a^ 

*  Euclid  7.  5.  t  Eiic.  9.  o. 


RATIO.  177 

Hence  a  recijfrocal  ratio  is  expressed  by  inverting  the  frac- 
tion which  expresses  the  direct  ratio  ;  or,  when  the  notation 
is  by  points,  by  inverting  the  order  of  the  terms. 

Thus  a  is  to  6,  inversely,  as  b  to  a, 

352.  Compound  ratio  is  the  ratio  of  the  products  of 
the  corresponding  terms  of  two  or  more  simple  ratios.^ 
Thus  the  ratio  of  6  :  3,  is  2 

And  the  ratio  of  1 2 :  4,  is  3 


The  ratio  compounded  of  these  is  72 ;  12=6. 

Here  the  compound  ratio  is  obtained  by  multiplying  to- 
gether the  two  antecedents,  and  also  the  two  consequents, 
of  the  simple  ratios. 

iSo  the  ratio  compounded. 

Of  the  ratio  of  a:b 

And  the  ratio  of  c:d 

An<d  the  ratio  of  h  ly 

Is  the  ratio  of  ach  :  hdy=^-ri-. 

Compound  ratio  is  not  different  in  its  nature  from  any  oth- 
er ratio.  The  term  is  used,  to  denote  the  origin  of  the  ra- 
tio, in  particular  cases. 

Cor.  The  compound  ratio  is  equal  to  the  product  of  the 
simple  ratios. 


The  ratio  of  a:6,  isT 

c 
The  ratio  of  c :  df,  is  -^ 

h 
The  ratio  of  h:y,is 


ach 
And  the  ratio  compounded  of  these  is  t^~,    which  is  the 

product  of  the  fractions  expressing  the  simple  ratios.     (Art 
155.) 

353.  If,  in  a  series  of  ratios,  the  consequent  of  each  pre- 
ceding couplet,  is  the  antecedent  pf  the  following  one,  the 

^  Sec  Note  I. 
24 


178  ALGEIJKA. 

ratio  of  the  first  antecedent  to  the  last  consequent,  is  equal  to 
ihat  which  is  compounded  of  all  the  intervening  ratios,^ 
Thus,  in  the  series  of  ratios      a :  b 

b:c 

Old 

d:h 
the  ratio  of  a :  h  is  equal  to  that  which  is  compounded  of 
the  ratios  of  a:b,  of  b  :  c,  of  c  :  d,  of  d :  h.     For  the  com- 

abcd     a 
pound  ratio,  by  the  last  article,  is  f~JL^~T'}  or  a  :  h»      (Art. 

145.) 

In  the  same  manner,  all  the  quantities  which  are  both  an- 
tecedents and  consequents  will  disappear  when  the  fractional 
product  is  reduced  to  its  lowest  terms,  and  will  leave  the 
compound  ratio  to  be  expressed  by  the  first  antecedent  and 
the  last  consequent. 

354.  A  particular  class  of  compound  ratios  is  produced, 
by  multiplying  a  simple  ratio  into  itself  or  into  another  equal 
ratio.  These  are  termed  duplicate,  triplicate,  quadruplicate, 
Sic.  according  to  the  number  of  multiplications. 

A  ratio  compounded  of  two  equal  ratios,  that  is,  the  square 
of  the  simple  ratio,  is  called  a  duplicate  ratio. 

One  compounded  of  three,  that  is,  the  ciiJbe  of  the  simple 
ratio,  is  called  triplicate,  Sic, 

In  a  similar  manner,  the  ratio  of  the  square  roots  of  two 
quantities,  is  called  a  subdujylicate  ratio  5  that  of  the  cube 
roots,  a  subtriplicate  ratio,  ($/'C. 

Thus  the  simple  ratio  of  a  to  6,  is  a :  6 
The  duplicate  ratio  of  a  to  b,  is  a^  :  b^ 
The  triplicate  ratio  of  a  to  Z»  is  a^  :  6^ 
The  subduplicate  ratio  of  a  to  b,  is  y/aiy/b 
The  subtriplicate  of  a  to  b,  is  ^a  :  ^/b,  &;c. 

The  terms  duplicate,  triplicate,  ^c,  ought  not  to  be  con- 
founded with  doubld,  triple,  tj*c.t 

The  ratio  of  6  to  2  is  6  :  2=3 

Double  this  ratio,  that  is,  twice  the  ratio  is     12:2=6  > 
Triple  the  ratio,  i.  e.  three  times  the  ratio,  is  18  :  2=9  5 

*  This  is  the  particular  case  of  compound  ratio  which  is  treated  of  in  tht 
5th  book  of  Euclid.     See  the  editions  of  Simsoa  and  Playfair. 
t  See  Note  .K. 


RATIO.  :^.    17$ 

But  the  duplicate  ratio,],  e.  the  square  of  the  ratio,  is  62 :  22  =  9  ^ 
And  the  triplicate  ratio,  i.  e.  the  cube  of  the  ratio,  is  6^:2^=27  > 

355.  That  quantities  may  have  a  ratio  to  each  other,  it  is 
necessary  that  they  should  be  so  far  of  the  same  nature,  as 
that  one  can  properly  he  said  to  he  either  equal  to,  or  great- 
er, or  less  than  the  other.  A  foot  has  a  ratio  to  an  inch,  for 
one  is  twelve  times  as  great  as  the  other.  But  it  can  not  be 
said  that  an  hour  is  either  longer  or  aliorter  than  a  rod  ;  or 
that  an  acre  is  greater  or  less  than  a  degree.  Still,  if  these 
quantities  are  expressed  by  numbers^  there  may  be  a  ratio 
between  the  numbers.  There  is  a  ratio  between  the  num- 
ber of  minutes  in  an  hour,  and  the  number  of  rods  in  a 
mile. 

356.  Having  attended  to  the  nature  of  ratios,  we  have 
next  to  consider  in  what  manner  they  will  be  affected,  by 
varying  one  or  both  of  the  terms  bet^veen  which  the  com- 
parison is  made.  It  must  be  kept  in  mind  that,  when  a  di- 
rect ratio  is  expressed  by  a  fraction,  the  antecedent  of  the 
couplet  is  always  the  numerator,  and  the  consequent,  the 
denominator.  It  will  be  easy,  then,  to  derive  from  the  prop- 
erties of  fractions,  the  changes  produced  in  ratios  by  varia- 
tions in  the  quantities  compared.  For  the  ratio  of  the  two 
quantities  is  the  same  as  the  value  of  the  fractions,  each  be- 
ing the  quotient  of  the  numerator  divided  by  the  denomina- 
tor. (Arts.  135,  346.)  Now  it  has  been  shown,  (Art.  137.) 
that  multiplying  the  numerator  of  a  fraction  by  any  quantity, 
is  multiplying  the  value  by  that  quantity  ;  and  that  dividing 
the  numerator  is  dividing  the  value.     Hence, 

357.  Multiplying  the  antecedent  of  a  couplet  by  any  quanti^ 
ty,  is  multiplying  the  ratio  by  that  quantity  ;  and  dividing  the 
antecedent  is  dividing  the  ratio. 

Thus  the  ratio  of      6  : 2  is  3 
And  the  ratio  of      24  : 2  is  12. 

Here  the  antecedent  and  the  ratio,  in  the  last  couplet,  are 
each  four  times  as  great  as  in  the  first. 

a 
The  ratio  of  a  :  6  is         -r- 

na 
And  the  ratio  of  na  :  Z>  is  -t". 


180  ALGEBRA. 

Cor.  With  a  given  consequent,  the  greater  the  antecedent^ 
the  greater  the  ratio  ;  and  on  the  other  hand,  the  greater  the 
ratio,  the  greater  the  antecedent.*     See  art.  137.  cor. 

358.  Multiplying  the  consequent  of  a  couplet  hy  any  quan- 
tity  is,  in  effect,  dividing  the  ratio  by  that  quantity  j  and  divi- 
ding the  consequent  is  multiplying  the  ratio.  For  multiply- 
ing the  denominator  of  a  fraction,  is  dividing  the  value  ;  and 
dividing  the  denominator  is  niultiplymg  the  value.  (Art. 
138.) 

Thus  the  ratio  of  12:2,  is  6 
And  the  ratio  of     12:4,  is  3. 

Here  the  consequent,  in  the  second  couplet,  is  twice  as 
great,  and  the  ratio  only  half  as  great,  as  in  the  first. 

a 
The  ratio  o(  axhis  ~r 

a 
And  the  ratio  of  a :  n&,  is  -r. 

Cor.  With  a  given  antecedent,  the  greater  the  consequent^ 
the  less  the  ratio  ;  and  the  greater  the  ratio,  the  less  the 
consequent.!     See  art.  138.  cor, 

359.  From  the  two  last  articles,  it  is  evident  that  multi- 
plying the  antecedent  of  a  couplet,  by  any  quantity,  will  have 
the  same  effect  on  the  ratio,  as  dividing  the  consequent,  by 
that  quantity  ;  and  dividing  the  antecedent  will  have  the  same 
effect  3iS  midtiplying  the  consequent.     See  art.  139. 

Thus  the  ratio  of  8  : 4,  is  2 

Mult,  the  antecedent  by  2,  the  ratio  of    16: 4,  is  4 
Divid.  the  consequent  by  2,  the  ratio  of     8:2,  is  4. 

Cor.  Any  factor  or  divisor  may  be  transferred,  from  the 
antecedent  of  a  couplet  to  the  consequent,  or  from  the  con- 
sequent to  the  antecedent,  without  altering  the  ratio. 

It  must  be  observed  that,  when  a  factor  is  thus  transferred 
from  one  term  to  the  other,  it  becomes  a  divisor  ;  and  when 
a  divisor  is  transferred,  it  becomes  a  factor. 

Transferring  the  factor  3,         6:f  =  25 

*  Euclid  8  and  10.  5.     The  first  part  of  the  propositions, 
t  Euclid  8  and  10.  5.     The  last  part  of  the  propositions. 


RATIO.  181 

ma         ma  ma  ^ 

The  ratio  of  J'^^^'^^^W  \ 

ma  \ 

Transferring  y,  ma  :  hy=^7iia-^  ^^y^'h^  f 

hy  hy      ma    I 

Transferingm,  ''''m^'^'^lii'^h^   J 

360.  It  is  farther  evident,  from  arts.  357  and  358,  that  iw 

THE  ANTECEDENT  AND  CONSEQUENT  BE  BOTH  MULTIPLIED,  OR 
BOTH  DIVIDED,  BY  THE  SAME  QUANTITY,  THE  RATIO  WILL  NOT 

BE  ALTERED.*     Sco  art-  140. 

Thus  the  ratio  of  8:4=2^ 

Mult,  both  terms  by  2,  16  :8=2>  the  same  ratio. 

Divid.  both  terms  by  2,  4: 2==2) 

a  '\ 

The  ratio  of  aib^^  I 

ma      a     \ 
Multiplying  both  terms  by  vi,  maimh—~i==-r 

a  ,h      an      a 
Dividing  both  terms  by  n,  -.-=-^=y  j 

Cor.  1 .  The  ratio  of  two  fractions  which  have  a  common 
denominator,  is  the  same  as  the  ratio  of  their  numerators, 

a     b 
The  ratio  of  — :  -^,  is  the  same  as  that  of  a  :b» 
n     n ' 

Cor.  2.  The  direct  ratio  of  two  fractions  which  have  a 
common  numerator,  is  the  same  as  the  reciprocal  ratio  of 
their  denominators, 

a      a  11 

Thus  the  ratio  of  —  ;  — ,  is  the  same  as  — 7 :  —  or  mm, 
m     n^  m     n 

361.  From  the  last  article,  it  will  be  easy  to  determine 
the  ratio  of  any  two  fractions.  If  each  term  be  multiplied 
by  the  two  denominators,  the  ratio  will  be  assigned  in  inte- 
gral expressions.     Thus,  multiplying  the  terms  of  the  coup- 

a     c  abd   bed 

let  -j-  :—j  by  bd,  we  have  ~r~ :  ~t~,  which  becomes  ad :  be,  by 

cancelling  equal  quantities  from  the  numerators  and  denom- 
inators. 

*  Euclid  15.  5. 


Ig2  ALGEBRA. 

361.6.  A  ratio  of  greater  inequality^  compounded  with  an- 
other ratio,  increases  it. 

Let  the  ratio  of  greater  inequaUty  he  that  of      1  +w:  1 
And  any  given  ratio,  that  of  a\h 

Tlie  ratio  compounded  of  these,  (Art.  352.)  is  a-\-na'.h 
Which  is  greater  than  that  part  of  a\h  (Art.  366,  cor.) 

But  a  ratio  of  lesser  inequality^  compounded  with  anoth- 
er ratio,  diminishes  it. 

Let  the  ratio  of  lesser  inequaUty  be  that  of       1  —w  :  1 
And  any  given  ratio,  that  of  a :  b 

The  ratio  compounded  of  these  is  a--na:h 

Which  is  less  than  that  of  a  :  5. 

362.  If  to  or  from  the  terms  of  any  couplet^  there  he  ad- 
ded or  SUBTRACTED  two  Other  quantities  having  the  same  ra- 
tio, the  sums  or  remainders  will  also  have  the  same  ratio,^ 
Let  the  ratio  of  aihl 

Be  the  same  as  that  of  cd^ 

Then  tlue  ratio  of  the  sum  of  the  antecedents,  to  the  sum 
of  the  consequents,  viz.  of  a-f  c  to  6-ff?,  is  also  the  same. 

a-^-c      c       a 
ThatiSjq:^=-;^=-j-. 

Demonstration. 

a       c 

1.  By  supposition,  ~b^~d 

2.  Multiplying  by  b  and  J,  adr=bc 

3.  Adding  cd  to  both  sides  ad-\-cdf=zbc-\-cd 

hc-\-cd 

4.  Dividing  by  J,  a-\-c=     ^ 

a-j-c      c       a 

5.  Dividing  by  6-f  <Z,  6+^='^'=T- 

The  ratio  of  the  difference  of  the  antecedents,  to  the  dif- 
ference of  the  consequents,  is  also  the  same. 
a—c      c       a    *> 

*  Euclid  5  and  6.  5. 


RATIO.  183 

Demonstraciion. 


c 


1.  By  supposition,  as  before,  17~~1 

2.  Multiplying  by  h  and  J,  ad=bc 

3.  Subtracting  cd  from  both  sides,  ad^cd=bc-'cd 

TA-  .  T      ,      t  hc—cd 

4.  Dividing  by  d,  a—c= — ^ — 


5.  Dividing  hy  b-^d 


a  —  c      c       a 


h-^d'^  d~~  b' 

Thus  the  ratio  of  15  :  5  is  3  ) 

And  the  ratio  of  9 :  3  is  3  S 

Then  adding  and  subtracting  the  terms  of  the  two  couplets. 

The  ratio  of  15  +  9:  5+3  is  3> 

And  the  ratio  of  15— 9:5— Sis  35 

Here  the  terms  of  only  hoo  couplets  have  been  added  to- 
gether. But  the  proof  may  be  extended  to  any  number  of 
couplets,  where  the  ratios  are  equal.  For,  by  the  addition 
of  the  two  first,  a  new  couplet  is  formed,  to  which,  upon  the 
same  principle,  a  third  may  be  added,  a  fourth,  &:c.     Hence, 

363.  If,  in  several  couplets,  the  ratios  are  equal,  the  sum 

OF  ALL  THE  ANTECEDENTS  HAS  THE  SAME  RATIO  TO  THE  SUM 
OF  ALL  THE  CONSEQUENTS,  WHICH  ANY  ONE  OP  THE  ANTECE- 
BENTS  HAS,  TO  ITS  CONSEQUENT.* 

fl2:6=2 
Thus  the  ratio  ^  ^8-4=2 
I,  6:3=2 
Therefore  the  ratio  of  (12+10+8  +  6):(6  +  5+4+3)=.2. 

363.6.  A  ratio  of  greater  inequality  is  diminished^  by  ad- 
ding the  same  quantity  to  both  the  terms. 

a+d 

Let  the  given  ratio  be  that  of  ^-M-\-h  :  a  ov  

a 

a  +  6+a2 
Adding  X  to  both  terms,  it  becomes  (?+5+^:fl+i2;or  ~~rTr~ 


184  ALGEBRA. 

Reducing  them  to  a  common  denominator, 

a^-\-ab'\-ax-\-bx 


The  first  becomes 
And  the  latter 


a^-{-ab-\-ax 


As  the  latter  numerator  is  manifestly  less  than  the  other^ 
the  ratio  must  be  less.     (Art.  356.  cor.) 

But  a  ratio  of  lesser  inequality  is  increased^  by  adding  the 
same  quantity  to  both  terms. 

a— 5 
Let  the  giyen  ratio  be  that  of  a—6: «,  or 

a  —  b-\-x 
Adding  J^  to  both  terms,  it  becomes  a^b-\-a;:a-{-x  or -r- 

Reducing  them  to  a  common  denoninator. 


The  first  becomes 
And  the  latter. 


a{a'^x) 

As  the  latter  numerator  is  greater  than  the  other,  the  ra- 
tio is  greater. 

If  the  same  quantity,  instead  of  being  added,  is  subtrac- 
ted from  both  terms,  it  is  evident  the  effect  upon  the  ratio 
must  be  reversed. 

Examples, 

1 .  Which  is  the  greatest,  the  ratio  of  11 : 9,  or  that  of 
44  : 35  ?  . 

2.  Which  is  the  greatest,  the  ratio  of  a -1-3  :J^  a,  or  that 
of  2a+7:|a? 

3.  If  the  antecedent  of  a  couplet  be  Q5,  and  the  ratio  13, 
what  is  the  consequent  ? 

4.  If  the  consequent  of  a  couplet  be  7,  and  the  ratio  IZ, 
what  is  the  antecedent  ? 

5.  "VYhat  is  the  ratio  compounded  of  the  ratios  of  3:7, 
and  2a  :  56,  and  7:c  + 1  :  3?/  —  2  ? 

6.  What  is  the  ratio  compounded  of  x-\-y :  6,  and 
x-^y ;  a  +  6,  and  a-\-b  :  h  ?  Ans,  x^  --^*  ;  bh» 

*  Euclid  1  and  12,  5. 


PJIOPORTION.  135 

7.  If  the  ratios  of  5x-\-7  :2op-3,  anda?4-2:ia;4-3  be  conu 
pounded,  will  they  produce  a  ratio  of  greater  inequality,  oi^ 
of  lesser  inequality  ? 

Ans.  A  ratio  of  greater  inequality. 

8.  What  is  the  ratio  compounded  of  jpH-^  *•  «?  and  x-^y  \h, 
and  h  : .^  Ans.  A  ratio  of  equality. 

9.  What  is  the  ratio  compounded  of  7:5,  and  the  dupli- 
GJate  ratio  of  4 :  9,  and  the  triplicate  ratio  of  3  : 2  ? 

Ans.  14:15. 

10.  What  is  the  ratio  compounded  of  3  ;  7,  and  the  tripli*- 
Gate  ratio  of  ^  :y,  and  the  subduplicate  ratio  of  49  :  9  ? 

Ans.  x^  :?/*. 

proportion! 

363.  An  accurate  and  famiUar  acquaintance  with  the  doc- 
trine of  ratios,  is  necessary  to  a  ready  understanding  of  the 
principles  of  proportion,  one  of  the  most  important  of  all  the 
branches  of  the  mathematics.  In  considering  ratios,  we 
compare  two  quantities^  for  the  purpose  of  finding  either 
their  difference,  or  the  quotient  of  the  one  divided  by  the 
other.  But  in  proportion,  the  comparison  is  between  two 
ratios.  And  this  comparison  is  limited  to  such  ratios  as  are 
equal.  We  do  not  inquire  how  much  one  ratio  is  greater  or 
less  than  another,  but  whether  they  are  the  same.  Thus  the 
numbers  12,  6,  8,  4,  are  said  to  be  proportional,  because  the 
ratio  of  12  :  6  is  the  same  as  that  of  8  :  4. 

364.  Proportion,  then,  is  an  equality  of  ratios.  It  is  ei- 
ther arithmetical  or  geometrical.  Arithmetical  proportion  is 
an  equality  of  arithmetical  ratios,  and  geometrical  propor- 
tion is  an  equality  of  geometrical  ratios.*  Thus  the  num- 
bers 6,  4,  10,  8,  are  in  arithmetical  proportion,  because  the 
difference  between  6  and  4  is  the  same  as  the  difference  be- 
tween 10  and  8.  And  the  numbers  6,  2,  12,  4,  are  in  geo- 
metrical  proportion,  because  the  ^/wohewf  of  6  divided  by  2  is 
the  same,  as  the  quotient  of  12  divided  by  4. 

365.  Care  must  be  taken  not  to  confound  proportion  with 
ratio.  This  caution  is  the  more  necessary,  as  in  conmion 
discourse,  Uie  two  terms  are  used  indiscriminately,  or  rather. 

*  See  Note  L. 
25 


18ti 


ALGEBRA. 


proportion  is  used  for  both.  The  expenses  of  one  man  are 
said  to  bear  a  greater  proportion  to  his  income,  than  those  of 
another.  But  according  to  tlie  definition  which  has  just 
been  given,  one  proportion  is  neither  greater  nor  less  than 
another.  For  equality  does  not  admit  of  degrees.  One  ra- 
tio may  be  greater  or  less  than  another.  The  ratio  of  12:2 
is  greater  than  that  of  G  :  2,  and  less  than  that  of  20 :  2.  But 
these  differences  are  not  applicable  to  proportion,  when  the 
term  is  used  in  its  technical  sense.  The  loose  signification 
which  is  so  frequently  attached  to  this  word,  may  be  proper 
enough  in  familiar  language  :  For  it  is  sanctioned  by  gener- 
al usage.  But,  for  scientific  purposes,  the  distinction  be- 
tween proportion  and  ratio,  should  be  clearly  drawn,  and 
cautiously  observed. 

366.  The  equality  between  two  ratios,  as  has  been  stated, 
is  called  proportion.  The  w^ord  is  sometimes  applied  also  to 
the  series  of  terms  among  which  this  equality  of  ratios  ex- 
ists. Thus  the  two  couplets  15:5  and  6  :  2  are,  when  taken 
together,  called  a  proportion. 

.  367.  Proportion  may  be  expressed,  either  by  the  common 
sign  of  equality,  or  by  four  points  between  the  tw^o  couplets, 

C8*'6  =  4-2,  or  8"6::4*'2>  are  arithmetical 

^^  (a  •*b=^c  "  d,  or  a  "  h  ::  c*  d)  proportions. 

C  1 2 :  6  =  8  : 4,  or  1 2  :  6  : :  8  :  4  >  are  geometrical 

^      ^    a  :b  =  d:h,  or    a:b::d  :h)  proportions. 

The  latter  is  read,  '  the  ratio  of  «  to  6  equals  the  ratio  of 
^  to  A  ;'  or  more  concisely,  '  a  is  to  b,  as  d  to  A.' 

368.  The  first  and  last  terms  are  called  the  extremes,  and 
the  other  two  the  means.  Homologous  terms  are  either  the 
two  antecedents  or  the  two  consequents.  Analogous  termg 
are  the  antecedent  and  consequent  of  the  same  couplet. 

369.  As  the  ratios  are  equal,  it  is  manifestly  immaterial 
which  of  the  tw^o  couplets  is  placed  first. 

.      a      c  c     a 

If  a:b  ::c:  J,  then  c:d::a:b.     For  if  -r  ='^  then  "3"= T» 

370.  The  number  of  terms  must  be,  at  least,  four.  For 
the  equality  is  between  the  ratios  of  two  couplets  ;  and  each 
couplet  must  have  an  antecedent  and  a  consequent.  There 
may  be  a  proportion,  however,  among  three  quantities.     For 


PROPORTION,.  187 

oBe  of  the  quantities  may  be  repeated^  so  as  to  form  two 
terms.  In  this  case,  the  quantity  repeated  is  called  the  mid- 
dle term,  or  a  mean  proportional  between  the  two  other  quan- 
tities, especially  if  the  proportion  is  geometrical. 

Thus  the  numbers  8,  4,  2,  are  proportional.  That  is, 
8  : 4  : :  4  :  2.  Here  4  is  both  the  consequent  in  the  first  coup- 
let, and  the  antecedent  in  the  last.  It  is  therefore  a  mean 
proportional  between  8  and  2. 

The  last  term  is  called  a  thii-d  proportional  to  the  two 
other  quantities.     Thus  2  is  a  third  proportional  to  8  and  4. 

371.  Inverse  or  reciprocal  proportion  is  an  equality  be- 
tween a  direct  ratio  and  a  reciprocal  ratio. 

Thus  4  :  2  :  :  I :  ^  ;  that  is,  4  is  to  2,  reciprocally^  as  3  to  6. 
Sometimes  also,  the  order  of  the  terms  in  one  of  the  coup- 
lets is  inverted,  without  wTiting  them  in  the  form  of  a  frac- 
tion.    (Art.  351.) 

Thus  4  :  2 :  :  3  :  6  inversely.  In  this  case,  the  first  term  is 
to  the  second^  as  the  fourth  to  the  third;  that  is,  the  first  di- 
vided by  the  second,  is  equal  to  the  fourth  divided  by  the 
third. 

372.  When  there  is  a  series  of  quantities,  such  that  the 
ratios  of  the  first  to  the  second,  of  the  second  to  the  third, 
of  the  third  to  the  fourth,  &;c.  arc  all  equal ;  tiie  quantities 
are  said  to  be  in  continued  proportion,  Tlie  consequent  of 
each  preceding  ratio  is,  then,  the  antecedent  of  the  follow- 
ing one.  Continued  proportion  is  also  called  progression, 
a«  will  be  seen  in  a  following  section. 

Thus  the  numbers  10,  8,  6,  4,  2,  are  in  continued  arithne- 
tical  proportion.     For  10  —  8  =  8  —  6=6  —  4=4—2. 

The  numbers  64,  32,  16,  8,  4,  are  in  continued  geometri- 
cal proportion.     For  64  :  32  : :  32  :  1 6  : :  16  :  8  :  :  8  :  4. 

If  a,  6,  c,  J,  A,  (Sic.  are  in  continued  geometrical  propor- 
tion ;  then  a  :b  ::b:c::c  :d  ::  d:h,  Sic, 

One  case  of  continued  proportion  is  that  of  three  propor- 
tional quantities.     (Art.  370.) 

373  As  an  arithmetical  proportion  is,  ^generally,  nothing 
more  than  a  very  simple  equation,  it  is  scarcely  necessary  to 
give  the  subject  a  separate  consideration. 

The  proportion  a'  *b::c"d 

Is  the  same  as  the  equation  fi-^h:=s.c^d> 


188  ALGEBRA. 

It  will  be  proper,  however,  to  observe  that,  if  four  quan- 
tities are  in  arithmetical  proportion,  the  sum  of  the  extremes 
is  equal  to  the  sum  of  the  means. 

Thus  if  «  •  •  5: : /t  ••  «i,  then  a-{-m^h-\-h 

For  by  supposition,  a—h—h—m 

And  transposing  —h  and  —m  a-\-m=.h-\-h 

S^o  in  the  proportion,  12  •  •  10::  1 1  '•9, we  have  124-9= 104 11. 

Again,  if  three  quantities  are  in  arithmetical  proportion^ 
the  sum  o(  the  extremts  is  equal  to  double  the  mean. 

If  fl  .  .6  :  :6. .  c,  then  a  —  bs=b  —  c 

And  transposing  —b  and  — c,  a-\-c=2b, 

GEOMETRICAL  PROPORTION. 

374.  But  if  four  quantities  are  in  geometrical  proportion, 
the  PRODUCT  of  the  extremes  is  equal  to  the  product  of  the 
means. 

If  a  :  6  : :  c  :  J,  adz=zhc 

a       c 
For  by  supposition,  (Arts.  346,  364.)         T^"^ 

abd     cbd 
Multiplying  by  bd,  (Ax.  3.)  — 7-=-^- 

Reducing  the  fractions,  ad-==-bc 

Thus  12:8::  15:10,  therefore  12x10=8x15.. 

Cor.  Any  factor  may  be  transferred  from  one  mean  to 
the  other,  or  from  one  extreme  to  the  other,  without  affec- 
ting the  proportion.  If  a  :  mb  :  x  :  1/,  then  a:b  ::  mx  : y. 
For  the  product  of  the  means  is,  in  both  cases,  the  same. 
And  \i  naxb  wx'.y,  then  a  :  6  : :  ^  :  ny, 

375.  On  the  other  hand,  if  the  product  of  two  quantities 
is  equal  to  the  product  of  two  others,  the  four  quantities  will 
form  a  proportion,  when  they  are  so  arranged,  that  those  on 
one  side  of  the  equation  shall  constitute  the  means,  and 
those  on  the  other  side,  the  extremes. 


If  my=7iA,  then  m  :n:  :h:  y,  that  is 
For  by  dividing  my=nh  by  ny.  wc  have 


And  reducing  the  fractions, 


m      h 

n  """  y 
my     nh 

7iy'~ny 


PROPORTION.  189 

Cor.  The  same  must  be  true  of  (my  factors  which  form 
the  two  sides  of  an  equation. 

If  (a-i-b) X c={d—m) X y,  then  a+b:  d—m::y:c. 

376.  If  three  quantities  are  proportional,  the  product  of 
the  extremes  is  equal  to  the  square  of  the  mean.  For  this 
mean  proportional  is,  at  the  same  time,  the  consequent  of 
the  first  couplet,  and  the  antecedent  of  the  last.  (Art  370.) 
It  is  therefore  to  be  multiplied  into  itself,  that  is,  it  is  to  be 
-squared* 

M  a:h  ::b:c,  then  mult,  extremes  and  means,  ac=b^. 

Hence,  a  mean  proportional  between  two  quantities  may 
be  found,  by  extracting  the  square  root  of  their  product. 
If  a  :  a: :  :  07 :  c,  then  x^  =aCj  and  x=  Vac,     (Art.  297.) 

377.  It  follows,  from  art.  374,  that  in  a  proportion,  either 
extreme  is  equal  to  the  product  of  the  means,  divided  by 
the  other  extreme ;  and  either  of  the  means  is  equal  to  the 
|)roduct  of  the  extremes,  divided  by  the  other  mean. 

^»  U  a:b::c:d,  then  ad=bc 

be 

2.  Dividing  by  d,  a=-j 

ad 

3.  Dividing  the  first  by  c,  o  =  — 

c 

ad 

4.  Dividing  it  by  b,  ^^17 

be 

5.  Dividing  it  by  a,  ^^~a  '  *^^^  ^^'  *^^  fourth 
term  is  equal  to  the  product  of  the  second  and  third  divided 
by  the  first. 

On  this  principle  is  founded  the  rule  of  simple  proportion 
in  arithmetic,  commonly  called  the  Rule  of  Three,  Three 
numbers  are  given  to  find  a  fourth,  which  is  obtained  by 
multiplying  together  the  second  and  third,  and  dividing  by 
the  first. 

378.  The  propositions  respecting  the  products  of  the 
means,  and  of  the  extremes,  furnish  a  very  simple  and  con- 
venient criterion  for  determining  whether  any  four  quanti- 
ties are  proportional.  We  have  only  to  multiply  the  means 
together,  and  also  the  extremes.  If  the  two  products  are 
equal,  the  quantities  are  proportional.  If  the  two  products 
are  not  equal,  the  quantities  are  not  proportional, 


190  ALGEBRA. 

379.  In  malheinatical  investigations,  when  the  relations  of 
several  quantities  are  given,  they  are  frequently  stated  in  the 
form  of  a  proportion.  But  it  is  commonly  necessary  that 
this  first  proportion  should  pass  through  a  number  of  trans- 
formations, before  it  brings  out  distinctly  the  unknown  quan- 
tity, or  the  proposition  which  vre  wish  to  demonstrate.  It 
may  undeigo  any  change  which  will  not  affect  the  equality 
of  the  ratios ;  or  which  will  leave  the  product  of  the  means 
equal  to  the  product  of  the  extremes. 

It  is  evident,  in  the  first  place,  that  any  alteration  in  the 
arrangement,  which  will  not  affect  the  equality  of  tliese  two 
products,  Avill  not  destroy  the  proportion.  Thus,  if  a :  /; : :  c :  d, 
the  order  of  these  four  quantities  may  be  varied,  in  anyway 
which  will  leave  ad^bc.     Hence, 

380.  If  four  quantities  are  proportional,  the  order  of  the 

MEANS,  OR  OF  THE  EXTREMES,  OR  OF  THE  TERMS  OF  BOTH 
COUPLETS,  MAY  BE  INVERTED,  WITHOUT  DESTROYING  THE  PRO- 
PORTION. 

If  a:  b  ::  c  :  d^    . 

And    IS:  8::  6:4  5  ' 

1 .  Inverting  the  means,* 

a:  c::b  :d}  .y   ,    ■     S  The Jirst,  is  to  the  third, 
12 : 6  : :  8  :  4  >  *^'  {  As  the  second,  to  the  fourth, 

In  other  words,  the  ratio  of  the  antecedents  is  equal  to  the 
ratio  of  the  consequents. 

This  inversion  of  the  means  is  frequently  referred  to  by 
geometers  under  the  name  of  Alternation,^ 

2.  Inverting  the  extremes, 

d:b  ::  c:  a    ")  .y    ^  -      ^  The  fourth,  is  to  the  second, 
4  : 8  : :  6  : 1 2  S  ^^'    i  As  the  third,  to  the  first. 

3.  Inverting  the  terms  of  each  couplet, 

h:  a  : :  J :  c  C  .1    ,  •      \  The  second,  is  to  the  first, 
8  :  1 2 : :  4  : 6  (  ^^'   \  As  the  fourth,  to  the  third. 

This  is  technically  called  Inversion, 

Each  of  these  may  also  be  varied,  by  changing  the  order 
»f  the  tico  couplets,     (Art.  369.) 

Cor.  The  order  of  the  whole  proportion  may  be  inverted.. 

If  a:b::c:d,  then  d:c::b:a' 

*  See  Noie  M.        +  Euclid  16.  5. 


PROPORTION.  1^1 

In  each  of  these  cases,  it  will  be  at  once  seen  that,  by 
taking  the  products  of  the  means,  and  of  the  extremes,  we 
have  ad— he,  and  12x4  =  8x6. 

If  the  terms  of  only  one  of  the  couplets  are  inverted,  the 
proportion  becomes  reciprocal,     (Art.  371.) 

If  a:b::c:d,  then  a  is  to  b,  reciprocally,  as  d  to  c. 

381.  A  difference  of  arrangement  is  not  the  onbj  alteration 
which  we  have  occasion  to  produce,  in  the  terms  of  a  pro- 
portion. It  is  frequently  necessary  to  multiply,  divide,  in- 
volve, &;c.  In  all  cases,  the  art  of  conducting  the  investiga- 
tion consists  in  so  ordering  the  several  changes,  as  to  main- 
tain a  constant  equality,  between  the  ratio  of  the  two  first 
terms,  and  that  of  the  two  last.  As  in  resolving  an  equa- 
tion, we  must  see  that  the  sides  remain  equal ;  so  in  varying 
a  proportion,  the  equality  of  the  ratios  must  be  preserved. 
And  this  is  effected,  either  by  keeping  the  ratios  the  same^ 
while  the  terms  are  altered  ;  or  by  increasing  or  diminishing 
one  of  the  ratios,  as  rtmch  as  the  other.  Most  of  the  suc- 
ceeding proofs  are  intended  to  bring  this  principle  distinctly 
into  view,  and  to  make  it  familiar.  Some  of  the  proposi- 
tions miglit  be  demonstrated,  in  a  more  simple  manner,  per- 
haps, by  multiplying  the  extremes  and  means.  But  thii 
would  not  give  so  clear  a  view  of  the  nature  of  the  several 
changes  in  the  proportions. 

It  has  been  shown  that,  if  ooth  the  terms  of  a  couplet  be 
multiplied  or  divided  by  the  same  quantity,  the  ratio  will  re- 
main the  same  ;  (Art.  360.)  that  multiplying  the  antecedent 
is,  in  effect,  multiplying  the  ratio,  and  dividing  the  antece- 
dent, is  dividing  the  ratio  ;  (Art.  357.)  and  farther,  that 
multiplying  the  consequent  is,  in  effect,  dividing  the  ratio,  and 
dividing  the  consequent  is  multiplying  the  ratio.  (Art.  358.) 
As  the  ratios  in  a  proportion  are  equal,  if  they  are  both  mul- 
tiplied, or  both  divided,  by  the  same  quantity,  they  will  still 
be  equal.  (Ax.  3.)  One  will  be  increased  or  diminished  as 
much  as  the  other.     Hence, 

382.  if  four  quantities  are  proportional,  two  analogous, 

OR  TWO  HOMOLOGOUS  TERMS  MAY  BE  MULTIPLIED  OR  DIVIDED 
BY  THE  SAME  QUANTITY,  WITHOUT  DESTROYING  THE  PROPOR- 
TION. 

If  analogous  terms  be  multiplied  or  divided,  the  ratios  will 
not  be  altered.  (Art.  360.)  If  homologous  terms  be  multi- 
pUed  or  divided,  both  ratios  will  be  equally  increased  or  di- 
minished. (Arts,  357.  8.) 


L 


192  ALGEBRA. 

It  a:  b ::  c:  d^  then, 

1.  Multiplying  the  two  first  termSy  ma:mb::c:d 

2.  Multiplying  the  two  last  terms,  a:b  :imc:  ind 

3.  Multiplying  the  two  antecedents,*  maibiimcid 

4.  Multiplying  the  two  consequents,  a:mb::c:  md 

a     b 

5.  Dividinor  the  two  first  terms,  —  :~  izcid 

c      d 

6.  Dividing  the  two  last  terms.  a:b::~ :  — 

7.  Dividing  the  two  antecedents,  —:b::  —  :ct 

,..  .,.       ,  b  d 

8.  Dividing  the  two  consequents,  a:  —  : : c :  — 

Cor.  1.  Jill  the  terms  may  be  multiplied  or  divided  by 
the  same  quantity.! 

,  abed 

ma  imb  i:mc'.  md      —  :  —  : : —  :  — 

,mm      mm 

Cor.  2.  In  any  of  the  cases  in  this  article,  multiplication 
of  the  consequent  may  be  substituted  for  division  of  the  an- 
tecedent in  the  same  couplet,  and  division  of  the  consequent, 
for  multiplication  of  the  antecedent.     (Art.  359,  cor.) 

Thjm  \  ^i<^  •  ^  •  •  ^^ic  :d  f  ^\a:—::mc:d  /      \ma:b::c  :  — 

a  c         i  >^)  ^(j^ 

—  :b::  — :  d    \  a  f  a:  mb  ::  —  '.d\      /  —  :J;:c:  md. 

m  m         ^SV.  m      J      K  m 


383.  It  is  often  necessary,  not  only  to  alter  the  terms  of  a 
proportion,  and  to  vary  the  arrangement,  but  to  compare  one 
proportion  with  another.  From  this  comparison  will  fre- 
quently arise  a  nev)  proportion,  which  may  be  requisite  in 
solving  a  problem,  or  iii  carrying  forward  a  demonstration. 
One  of  the  most  important  cases  is  that  in  which  two  of  the 
terms  in  one  of  the  proportions  compared,  are  the  same  with 
two  in  the  other.  The  similar  terms  may  be  made  lo  disap- 
pear^ and  a  new  proportion  may  be  formed  of  the  four  re- 
maining terms.     For, 

*  Euclid  3.  5.  i   Euclid  4.  5. 


PROPORTION.  J  93 

384.  If  two  ratios  are  respectively  equal  to  a  third, 
they  are  equal  to  each  other.* 

This  is  nothing  more  than  the  Uth  axiom  appUed  to 
ratios. 

2.  If     a :  i  ; :  m :  n  >  .  1  7  7  ,     , 

7   >  then  a:  0  : :  G  :  a,  or  a :  c  ; :  6  ;  a. 
:n: :c : a  y  ' 


*  7  >  then  a:h  2^  c  :  J.t 


And  m 

Cor.  If  a:h  : :  m 
7n:n  -^  c 

For  if  the  ratio  of  m  :n  is  greater  than  that  of  c  ;  </,  it  is 
manifest  that  the  ratio  of  a  :  b,  which  is  equal  to  that  of  m :  72, 
is  also  greater  than  that  of  c  :  d. 

385.  In  these  instances,  the  terms  which  are  ahke,  in  the 
two  proportions,  are  the  two  Jirst  and  the  two  last.  But  this 
arrangement  is  not  essential.  The  order  of  the  terms  maj 
be  changed,  in  various  ways,  without  affecting  the  equality  of 
the  ratios. 

1.  The  similar  terms  may  be  the  two  antecedents,  or  the 
two  consequents,  in  each  proportion.     Thus, 

If      m :  a : ;  n  :  6  )  ,7        J  By  alternation,  m  :n:  :a:b 
And  m  :  c  :  :n  :d)  \  And  m  :n:  :c  : d 

Therefore  a:h::c  :  d,  or  a  :c::b  :  d,  by  the  last  article. 

2.  The  antecedents  in  one  of  the  proportions,  may  be  the 
same  as  the  consequents  in  the  other. 

If     m',aiin:bl.y^       (By  inver.  and  altern.  aib'.imin 
And  c',miid:n^  (By  alternation,  cidiimin 

Therefore  a  :  b,  he,  as  before. 

3.  Two  homologous  terms,  in  one  of  the  proportions,  may 
be  the  same,  as  two  aiuilogous  terms  in  the  other. 

If      a:m::b  :n}  .y^       (By  alternation,     a  :b::m:n 
And  c:d::m:n^  \  And  c:d::m:n 

Therefore  a  :  6,  &:c. 

All  these  are  instances  of  an  equality,  between  the  ratios 
in  one  proportion,  and  those  in  another.    In  geometry,  the 

♦^Euciidll.  5.  tEuelidlS.  a. 

26 


J34  ALGEBRA. 

proposition  to  which  they  belong  is  usually  cited  by  the 
words  "  ex  aequo,^'^  or  "  ex  aequalV*  The  second  case  in 
this  article  is  that  which,  in  its  form,  most  obviously  answers 
to  the  explanation  m  Euchd.  But  they  are  all  upon  the 
same  principle^  and  are  frequently  referred  to,  without  dis- 
crimination. 

386.  Any  number  of  proportions  may  be  compared,  in 
the  same  manner,  if  the  two  first  or  the  two  last  terms  in 
each  preceding  proportion,  are  the  same  with  the  two  first 
or  the  two  last  in  the  following  one,* 


Thus  if 

And 

And  hi  I: 

And  m 


a  :  b  : :  c  '  d^ 

:l::m:ni  ^ 

liiiix  lyj 


That,  is,  the  two  first  terms  of  the  first  proportion  have 
the  same  ratio,  as  the  two  last  terms  of  the  last  proportion. 
For  it  is  manifest  that  the  ratio  of  all  the  couplets  is  the 
same. 

And  if  the  terms  do  not  stand  in  the  same  order  as  here, 
yet  if  they  can  be  reduced  to  this  form,  the  same  principle  is 
applicable, 

Thus  a  a:  c::h  :d'^ 

And        child:  l{  .1       ■,       ,,         ,. 
i    J        ,  ,       >  then  by  alternation- 

And       himiil  mf  -^ 

And       mix:  iniyj 

Therefore  aibiix  ly,  as  before. 

In  all  the  examples  in  this,  and  the  preceding  articles,  the 
two  terms  in  one  proportion  which  have  equals  in  another, 
are  neither  the  two  means,  nor  the  two  extremes,  but  one  of 
the  means,  and  one  of  the  extremes  ;  and  the  resulting  pro- 
portion is  uniformly  direct, 

387.  But  if  the  two  means,  or  the  two  extremes,  in  one 
proportion,  be  the  same  with  the  means,  or  the  extremes,  in 
another,  the  four  remaining  terms  will  be  reciprocally  pro- 
porlional. 

If      aimiinib}^  1       1 

And   c:'/n::n:rf5tbentf:c::y  ;-j,or«;c::tZ;6. 

For  ab^mn )  ^^^.^^  3.^^  >^  Therefore  a6=:cJ, and a:c::d:b. 
And  cd=m7i  )  ^  ^ 

*  Euclid  22.  5. 


PROPORTION.  195 

In  this  example,  the  two  means  m  one  proportion,  are  like 
those  in  the  other.  But  the  principle  will  be  the  same,  if 
the  extremes,  are  alike,  or  if  the  extremes  in  one  proportion 
are  like  the  means  in  the  other. 


If      m:  a  ;:b  :n}  .,  ^^ i . , 

A    1  7       >  then  a :  c  : :  a :  0. 

And  m  :c  :  :d:n  S 

Or  if  « :  m  : :  n  :  6  >  ,,  77 

.J  1        >  then  a:  c::  a :  0, 

And    m :c  ::a  :  n  y 


The  proposition  in  geometry  which  apphes  to  this  case,  ia 
usually  cited  by  the  words   "  eoc  aequo  pcrturbaie,'^'^* 

388.  Another  way  in  which  the  terms  of  a  proportion  may 
be  varied,  is  by  addition  or  subtraction,  ^ 

If  to  or  from  two  analogous  or  two  homologous  terms 

OF  A  proportion,  TWO  OTHER  QUANTITIES  HAVING  THE  SAME 
RATIO  BE  ADDED  OR  SUBTRACTED.  THE  PROPORTION  WILL  BE 
PRESERVED*! 

For  a  ratio  is  not  altered,  by  adding  to  it,  or  subtracting 
from  it,  the  terms  of  another  equal  ratio.     (Art.  362.) 


If        a:b::c:d~) 
And  a  :b  : :m  :nS 


Then  by  adding  to,  or  subtracting  from  a  and  Z>,  the.  terms 
of  the  equal  ratio  m :  n,  we  have, 

a-{-7n:b-{-n:  :c:d  and  a—m:b—n  ::  c  :d. 

And  by  adding  and  subtracting  m  and  n,  to  and  from  c  and 
d  we  have, 

a  :b  : :  c+m  :  d+n,  and  a: b  : : c—m  : d—n. 

Here  the  addition  and  subtraction  are  to  and  from  analo- 
gous terms.  But  by  alternation,  (Art.  380,)  these  terms  will 
become  homologous,  and  we  shall  have, 

a+m  :  c  : :  6+?i :  d,  and  a—m :  c  :  :b—n  :  d. 

Cor.  1 .  This  addition  may,  evidently,  be  extended  to  any 
numberlof  equal  ratios.| 

C   c:d 

Thusif  a;6;;-<    ^'^ 
\  m:n 

i   oc:y 

Then  a:b  ::c-{-h-\-m'\-x  :d-\-l-{'n+y,' 

*  Euclid  23,  5.  tEucUd2,  5,  |  Euclid  2,  5.    Cor. 


196  ALGEBRA. 

Cor.  2.  If  a:6;:c:J7  .,  ,_    »         ,        j^ 

And    m:o  :  .'Jt.'a  > 

For  by  alternation  a  :c::h :  d')  there-  ^       a+m:  c-^n :  :h:d 
And  m:n::h:d)    fore    (  or  a+m:6.*:c4-w.'c?. 

'389.  From  the  last  article  it  is  evident  that  if,  in  any  pro- 
portion, the  terms  be  added  to,  or  subtracted  from  each  other, 
that  is, 

If  two  analogous  or  homologous  terms  be  added  to, 
OR  subtracted   from  the  two  others,  the  proportion 

WILL  BE  preserved. 

Thus,  if  a  ;  i&  ;  ;  c  ;  rf,    and  12  : 4  : ;  6  :  a,  then, 

1 .  Adding  the  two  last  terms,  to  the  two  first. 

a-\-c:h-\-d::a:h  12  + 6  ;  4  +  2; :  12  :4 

anda  +  c:6  +  J::c:J  12+6:4+2;:6  :2 

or  a-Jt-c:a::h-\-d:h  12+6  ;  12:  :4+2:  4 

anda  +  c:c::6  +  J:d'.  12+6  :  6  .•.•4+2:2. 

2.  Addiyig  the  two  antecedents,  to  the  two  consequents. 
a-\-h:h  ::c-ird:d  12  +  4:4  ::6  +  2:2 
a+&:a::c+J:c,  <$rc.  12  +  4  :  12::6  +  2:6,  c^c. 

This  is  called  Composition.^ 

3.  Subtracting  the  two  first  terms,  from  the  two  last, 

c—a:  a  :  :d—h  :  h 
c—a  :  c : :  d—b  :  d,  tj-c. 

4.  Subtracting  the  two  last  terms,  from  the  two  first. 

a  —  c  :b—d::a:bl 

a  —  c:b^d::c:d,(^c. 

5.  Subtracting  the  consequents,  from  the  antecedents. 

a  —  b:b::c—d:d 

a:  a  —  b  ::c:c—d,  <^c. 

The  alteration  expressed  by  the  last  of  these  forms  is  call- 
ed Conversion. 

6.  Subtracting  the  antecedents,  from  the  consequents. 

b  —  a:  a  : :  d—c:c 

b  :  b—a  ::  d  :  d—c,  &;c. 

*  Euclid  24,  5.  i  Euclid  18,  5.  $  Euclid  19,  5'. 


PROPORTION.  197 

7.  Adding  and  subtracting, 

a-^-b  :  a—b  ::c-{-d  :c  —  d. 

That  is,  the  sum  of  the  two  first  terms,  is  to  their  differ- 
ence, as  the  sum  of  the  two  last,  to  their  difference. 

Cor.  If  any  compound  quantities,  arranged  as  in  the  pre- 
ceding examples,  are  proportional,  the  simple  quantities  of 
which  they  are  compounded  are  proportional  also. 

Thus,  if  a-\-b  :b  ::c-{-d:d,  then  a:bi:c:d. 

This  is  called  Division,^ 

390.  If  the  corresponding  terms  of  two  or  more 
ranks  of  proportional  quantities  be  multiplied  togeth- 
er, the  products  will  be  proportional. 

This  is  compounding  ratios,  (Art.  352,)  or  compounding 
proportions.  It  should  be  distinguished  from  what  is  called 
composition,  which  is  an  addition  of  the  terms  of  a  ratio. 
(Art.  389.  2.) 

If         a:b::c:dl  ]2:4::6:2 


:nS 


And     h:l::m:n\  10:5::  8:4 


Then     ah:bl::cm:  dn,  120:20:  :  48  :  8. 

For,  from  the  nature  of  proportion,  the  two  ratios  in  the 
first  rank  are  equal,  and  also  the  ratios  in  the  second  rank. 
And  multiplying  the  corresponding  terms  is  multiplying  the 
ratios,  (Art.  352.  cor.)  that  is,  multiplying  equals  by  equals  ; 
CAx.  3.)  so  that  the  ratios  will  still  be  equal,  and  therefore 
the  four  products  must  be  proportional. 

The  same  proof  is  applicable  to  any  number  of  propor- 
tions. 

Ca:  6:  :  c  :  df 

li  ih:  liimxn 

(p:q::x:y 

Then  ahp  :blq::  cmx  :  dny. 
From  this  it  is  evident,  that  if  the  terms  of  a  proportion 
be  multiphed,  each  into  itself,  that  is,  if  they  be  raised  to  any 
power,  they  will  still  be  proportional. 

U  a  :b::c:d  2:  4: :6:  12 

a:b::c:d  2 : 4 : : 6 : 1 2 


Then  a^  :b^  ::c^  :d^  4:16::  36  :  144 

*  Euclid  17.  5.     See  Note  N. 


J93  ALGEBRA. 

Proportionals  will  also  be  obtained,  by  reversing  this  pro- 
cess, that  is,  by  extracting  the  roots  of  tiie  terms. 

If  « :  6  : :  c  :  Jy  then   ^a:  ^Jhi :  y/c :  df. 

For,  taking  the  product  of  extr.  and  means,  ad=hc 
And  extracting  both  sides,  ^ad=  ^hc 

That  is,  (Arts.  259,  375.)  ^a  :  y/h  :  :  ,/c  :  ^ct. 

Hence, 

391.  If  several  quantities  are  proportional,  their  like 

POWERS  OR  LIKE  ROOTS  ARE  PROPORTIOXAL.* 

If  a:b::c:  d 
Then  «" :  6" : :  c" :  d".  and  %/a  :  '^/b : :  '^/c :  !J/t?. 

And  !J/«« :  !;^&" : :  ^c"  ;  !J/ J",  that  is,  a«" ;  6«  : :  c"";  <^. 

392.  If  the  terms  in  one  rank  of  proportionals  be  divided 
by  the  corresponding  terms  in  another  rank,  the  quotients 
will  be  proportional. 

This  is  sometimes  called  the  resolution  of  ratios. 

If      a:h::cid\  12:6::  18:9) 

And  h:l:  :m;nj  6:2::    9:3> 

abed  ^^^l^A 

This  is  merely  reversing  the  process  in  art.  390,  and  may 
be  demonstrated  in  a  similar  manner. 

This  should  be  distinguished  from  what  geometers  call 
division,  which  is  a  subtraction  of  the  terms  of  a  ratio. 
(Art.  389.  cor.) 

When  proportions  are  compounded  by  multiplication,  it 
will  often  be  the  case,  that  the  same  factor  will  be  found  in 
two  analogous  or  two  homologous  terms. 

Thus  if   a:b::c:d} 
And        m:a::7i:  c ^ 


am  :  ab:  :  en:  cd 
Here  a  is  in  the  two  first  terms,  and  c  in  the  two  last. 
Dividing  by  these,  (Art.  382,)  the  proportion  becomes 
m  :  b  :  :  n  :  d.         Hence, 

*  It  must  not  be  inferred  from  this,  that"  quantities  have  the  same  ratio, 
as  their  Uke  powers  or  like  roots.     See  art.  354. 


PROPORTION.  199 

393.  In  compounding  proportions,  equal  factors,  or  divi- 
sors in  two  analogous  or  homologous  terms  may  be  rejected^ 

Ca:b::c:d  12:4::9:3 

If    h:h::d:l  4:8::3:6 

rh:m;:l:n  8:20::6:  15 


Then   a:m:ic:n  12:20::9:15 

This  rule  may  be  appUed  to  the  cases,'  to  which  the  terms 
^^ex  aequo,^^  and  "  ex  aequo  perturbate''^  refer.  See  arts.  383 
and  387.  One  of  the  methods  may  serve  to  verify  the 
other. 

394.  The  changes  which  may  be  made  in  proportions, 
without  disturbing  the  equahty  of  the  ratios,  are  so  numer- 
ous, that  they  would  become  burdensome  to  the  memory, 
if  they  were  not  reducible  to  a  few  general  principles.  They 
are  mostly  produced, 

1.  By  inverting  the  order  of  the  terms,  Art.  380. 

2.  By  midtiplying  or  dividing  by  the  same  quantity^  Art.  382. 

3.  By  comparing  proportions  which  have  like  terms,  Art. 
384,  5,  6,  7. 

4.  By  adding  or  subtracting  the  terms  of  equal  ratios,  Art* 
388,  9. 

5.  By  muUlplying  or  dividing  one  proportion  by  another, 
Art.  390,  2,  3. 

6.  By  involving  or  extracting  the  roots  of  the  terms,  Art.  391. 

395.  When  four  quantities  are  proportional,  if  the^r^^  be 
greater  than  the  second,  the  third  will  be  greater  than  the 
fourth  ;  if  equal,  equal  ;  if  less,  less. 

For,  the  ratios  of  the  two  couplets  being  the  same,  if  one 
is  a  ratio  of  equality,  the  other  is  also,  and  therefore  the  an- 
tecedent in  each  is  equal  to  its  consequent ;  (Art.  350.)  if 
one  is  a  ratio  of  greater  inequality,  the  other  is  also,  and 
therefore  the  antecedent  in  each  is  greater  than  its  conse- 
quent ;  and  if  one  is  a  ratio  of  lesser  inequality,  the  other  is  * 
also,  and  therefore  the  antecedent  in  each  is  less  than  its  con- 
sequent. 

Ca=:b,  c==d 

Let  a:b  ::  c  :  d  ;  then  if   <  a>  b,  c>  cT 

fa^b.c-^d. 


200  ALGEBRA. 

Cor.  1 .  If  the  first  be  greater  than  the  third,  the  secx>nd 
will  be  greater  than  the  fourth  ;  if  equal,  equal;  if  less, 
less.* 

For  by  alternation,  a:b  ::c:d  becomes  aiciib  :  d,  with- 
out any  alteration  of  the  quantities.  Therefore,  if  a=&,  c=(?, 
&:c.  as  before. 

Cor.  2.  If    a  :  w  : :  c  :  n  V    ,,        ./.         ,  704. 

i   A  ^    J.  ^}   then  it  a =6,  c=a,  &ct. 

For,  by  equality  of  ratios,  (Art.  385.  2.)  or  compounding 
ratios,  (Arts.  390,  393.) 

a:b::c:d.     Therefore,  if  a =6,  c=id,  &;c.  as  before. 

Cor.  3.  If     a:m::7t:d)    ,.       ./.        /         70     + 
Andm:6::c:«l   then  if  a=6,  c=d,  ^c.J 

For,  by  compounding  ratios,  (Arts.  390,  393.) 

a:b  :  :c:i  d.     Therefore,  if  a=b,  c=d,  &;c. 

395. &.  If  four  quantities  are  proportional,  their  reciprocals 
are  proportional ;  and  v.  v. 

1       1        1      1 

If  a:b  ::  c:  d.  then  —  :  t~  : :  —  :  ~t. 
'  a      Q        c      a 

For  in  each  of  these  proportions,  we  have,  by  reduction^ 
ad=bc. 

CONTINUED  PROPORTION. 

396.  When  quantities  are  in  continued  proportion,  all  the 
ratios  are  equal.    (Art,  372.)     If 

a  :  b  :  I  b  :c  :  :  c  :  d  ::  d  :  Cj 
the  ratio  of  a:b  is  the  same,  as  that  of  b  :  c,  of  c  :  d,  or  of 
d  :  e.     The  ratio  of  the  first  of  these  quantities  to  the  last, 
is  equal  to  the  product  of  all  the  intervening  ratios ;  (Art. 
353,)  that  is,  the  ratio  of  a  :  e  is  equal  to 

abed 

But  as  the  intervening  ratios  are  all  egual,  instead  of  mul- 
tiplying them  into  each  other,  we  may  multiply  any  one  of 
them  into  itself  ^  observing  to  make  the  number  of  factors 

*  EucHd  14.  5.  t  Euclid  20.  S.  t  EucUd  21.  5.- 


PROPORTION.  2ei 

equal  to  the  number  of  intervening  ratios.     Thus  the  ratio 
of  a  :  c,  in  the  example  just  given,  is  equal  to 
a       a       a       a      a* 

When  several  quantities  are  in  continued  proportion,  the 
number  of  couplets,  and  of  course,  the  number  of  ratios,  is 
9ne  le»s  than  the  number  of  quantities.  Thus  the  five  pro- 
portional quantities  a,  6,  c,  d,  e,  form  four  couplets  contain- 
ing four  ratios  ;  and  the  ratio  of  «  :  e  is  equal  to  the  ratio  of 
«*  :b*y  that  is,  the  ratio  of  the  fourth  power  of  the  first 
quantity,  to  the  fourth  power  of  the  second.     Hence, 

397.  If  three  quantities  are  proportional,  the  first  is  to  the 
third,  as  the  square  of  the  first,  to  the  square  of  the  second  ; 
or  as  the  square  of  the  second,  to  the  square  of  the  third. 
In  other  words,  the  first  has  to  the  third,  a  duplicate  ratio  of 
the  first  to  the  second.  And  conversely,  if^the  first  of  three 
quantities  is  to  the  third,  as  the  square  of  the  first  to  the 
square  of  the  second  ;  the  three  quantities  are  proportional. 

U  a  :b :  :b:  c,  then  a:c:  :a^  :b^.     Universally, 

398.  If  several  quantities  are  in  continued  proportion, 
the  ratio  of  the  first  to  the  last  is  equal  to  one  of  the  interven- 
ing ratios  raised  to  a  power  whose  index  is  one  less  than  the 
number  of  quantities. 

If  there  are /o2<r  proportionals  «,  6,  c,  J,  then  a:d::a^  :b^. 
If  there  are  five  a,  b,c,d,e,  aieiia*  :b*,  &c. 

399.  If  several  quantities  are  in  continued  proportion, 
they  will  be  proportional  when  the  order  of  the  whole  is  in- 
verted. This  has  already  been  proved,  with  respect  to  four 
proportional  quantities.  (Art.  380.  cor.)  It  may  be  extended 
to  any  number  of  quantities. 

Between  the  numbers,  64,  32,  16,  8,  4, 
The  ratios  are  2,    2,    2,  2, 

Between  the  same  inverted  4,  8,  16,  32,  64, 
The  ratios  are  |,    J,     ^,     i. 

So  if  the  order  of  any  proportional  quantities  be  invert- 
ed, the  ratios  in  one  series  will  be  the  reciprocals  of  those  in 
the  oiher.  For,  by  the  inversion,  each  antecedent  becomes 
a  consequent,  and  v,  v,  and  the  ratio  of  a  consequent  to  its 
antecedent  is  the  reciprocal  of  the  ratio  of  the  antecedent 
27 


I 


%Q(-2  ALGEBRA. 

to  the  consequent.     (Art.  351.)     That  the  reciprocals  of 
equal  quantities  are  themselves  equal,  is  evident  from  ax.  4. 

Examples,        f 

1.  Divide  the  number  49  into  two  such  parts,  that  the 
greater  increased  by  6,  may  be  to  the  less  diminished  by  1 1  ; 
as  9  to  2. 

Let  37=  the  greater,  and  49— :v=r the  less. 

By  the  conditions  proposed,  aj  +  6  :  38— r  : :  9  :  2 

Adding  terms,  (Art.  389,  2.)  a:+6;  44  ::  9  :  11 

Dividing  the  consequents,  (Art.  382,  8.)  a? +6  :  4  : :  9  :  1 

Multiplying  extremes  and  means,      a:  +  6  =  36.     And  a? =30. 

.  2.  What  number  is  that,  to  which  if  1,5,  and  1 3,  be  sev- 
erally added,  the  first  sum  shall  be  to  the  second,  as  the  sec- 
ond to  the  third  ? 

Let  a;=s  the  number  required. 
By  the  conditions,  x-^\  ix-\-5  i:x-\-5  ix-\-l6 

Subtracting  t^rms,  (Art.  389,  6.)  a;-f  1  ;  4  : :  a:-f5  :  8 
Therefore  8^+8=4x4-20.     And  :v=3. 

3.  Find  tw  o  numbers,  the  greater  of  which  shall  be  to  the 
less,  as  their  sum  to  42;  and  as  their  difference  to  6. 

Let  a;  and  y  =  the  numbers. 
By  the  conditions  xiy:'.x-\-y\A2 

And  X  :y::x—y  :  6 

By  equality  of  ratios  x+y  :42::  x—y  :  6 

Inverting  the  means  :^'-\-y  :  x—y  w  42i& 

Adding  and  subtracting  terms,  (Art.  389,  7,)   2a; :  2y:  :48  :  36 
Dividing  terms,  (Art.  382.)  ac :  y  :  :  4  :  3 

4w 
Therefore  3a;=4y     And  ^="3'- 

From  the  2d  proportion,  6;v=y  x  (a;— t/) 

Substituting  -  for  ^,  2/=--24.     And  ;c=32. 

4.  Divide  the  number  18  into  two  such  parts,  that  the 
squares  of  those  parts  may  be  in  the  proportion  of  25  to  l&. 

Let  x^  the  greater  part,  and  18  — a;=tbe  less. 


X  rtv^r  vii 

LXlVi^,                                                                 ^ynj 

a?*:(18-^)*  ::25:16 

•) 

x:  18  — X  ::5:  4 

or :  1 8  : :  5  :  9 

X  :   2  : :  5  :  1 

a:  =  10. 

By  the  conditions, 
Extracting,  (Art.  391.) 
Adding  terms, 
Dividing  terms. 
Therefore, 

5.  Divide  the  number  14  into  two  such  parts,  that  the 
quotient  of  the  greater  divided  by  the  less,  shall  be  to  the 
quotient  of  the  less  divided  by  the  greater,  as  16  to  9. 

Let  x=  the  greater  part,  and  14  — x=  the  less. 

X         14—^ 
By  the  conditions,  iTH^  '  — J~  : :  16  :  9 

Multiplying  terms,  ^2  .  (i4_;cj2  ; :  jg  ;  0 

Extracting,  ^  :  1 4— j?  :  :  4 :  3 

Adding  terms,  os :  1 4  : :  4  :  7 

Dividing  terms,  .^ :  2  :  :  4  :  1 
Therefore,  a? =8. 

6.  If  the  number  20  be  divided  into  two  parts,  which  are 
to  each  other  in  the  duplicate  ratio  of  3  to  1,  what  number 
is  a  mean  proportional  between  those  parts  ? 

Let  J^=  the  greater  part,  and  20— a;  =  the  less. 
By  the  conditions,  a;:20-x::3^  :l^  ::9  :  \ 

Adding  terms  .  a; :  20  : :  9  : 1 0 

Therefore  a;  =  18.     And  20— .r=: 2 

A  mean  propor.  between  18  and  2  (Art.  376.)  =\/2x  18=6. 

7.  There  are  two  numbers  whose  product  is  24,  and  the 
difference  of  their  cubes,  is  to  the  cube  of  their  difference, 
as  1 9  to  1 .     What  are  the  numbers  ? 

Let  :f  and  y  be  equal  to  the  two  numbers. 

1.  By  supposition  a:^'=24> 

2.  And  ^^«-3/3:Cr-y)3::19:1  i 

3.  Or,  (Art.  217.)         x^  -y^  :x^  -3x^'y-\-3xy''  -y^  :  :}9:l 

4.  Therefore,  (Art.  389,  5)       ^x^y-Sxy'' :  (^-y)^  : :  18  : 1 

5.  Dividing  by  x  -y  (Art.  382,  5)  ^xy  :  {^-yY  : :  1 8  :  1 

6.  Or,  as  Sry  =  3X24  =  72,  72  :  (^r-y)^  : ;  13  :  1 

7.  Multiplying  extremes  and  means,  (3?— y)2=4 

8.  Extracting,  -  a;— y  =  2    > 

9.  By  the  first  condition,  we  have  a:y  =  24  > 

Reducing  these  two  equations,  we  have  x^Q,  and^=4. 


204  ALGEBRA. 


8.  It  is  required  to  prove  that     a:x:: V2a—y  :  Vy 

on  supposition  that  {a-{-xy  :  (a—xY  : :  cc+y  :x—y* 

1.  Expanding,        a^ -{•2ax-\-x^  la^ —2ax-^x^  ::x-{-y:x—y 

2.  Adding  and  subtracting  terms,       2a^  +2a;2  :  4ax  : :  2a; :  % 

3.  Dividing  terms,  a^  +x^  :  2aa; :  :x:y 

4.  Transf.  the  factor  jc,  (Art.  374.  cor.)     a^-^-x''  :  2a :  ix'^iy 

5.  Inverting  the  means,  a^  +^^  :  x^  :  :2a:y 

6.  Subtracting  terms,  a^  :  x^  :  :  2a— y :  2/ 

7.  Extracting  a  :  a? ; :  V^a—y :  Vy* 

9.  It  is  required  to  prove  that  dx  =  cy,  if  a?  is  to  y  in  the 
triphcate  ratio  of  a:b,  and  a  :b  ::  Vc  +  jc  :  ^yd'^y• 

1.  Involving  terms.  a^  :  5^  : :  c-{-^ :  d-^y 

2.  By  the  first  supposition,  a^  :h^  ii^  ly 

3.  By  equahty  of  ratios,  C'\-x'.d+y'.'.xiy 

4.  Inverting  the  means,  c-f  a; :  ^  : :  </4-y  -y 

5.  Subtracting  terms,  c,  xi'.diy 

6.  Therefore,  dx^cy. 

10.  There  are  two  numbers  whose  product  is  135,  and  the 
difference  of  their  squares,  is  to  the  square  of  their  differ- 
ence, as  4  to  1 .     What  are  tlie  numbers  ? 

Ans.   15  and  9. 

11.  What  two  numbers  are  those,  whose  difference,  sum, 
g-nd  product,  are  as  the  numbers  2,  3,  and  5,  respectively  ? 

Ans.  10  and  2. 

1 2.  Divide  the  number  24  into  two  such  parts,  that  their 
product  shall  be  to  the  sum  of  their  squares,  as  3  to  10. 

Ans.  18  and  6. 

13.  In  a  mixture  of  rum  and  brandy,  the  difference  be- 
tween the  quantities  of  each,  is  to  the  quantity  of  brandy,  as 
100  is  to  the  number  of  gallons  of  rum  ;  and  the  same  dif- 
ference is  to  the  quantity  of  rum,  as  4  to  the  number  of  gal- 
lons of  brandy.     How  many  gallons  are  there  of  each  ? 

Ans.  25  of  rum,  and  5  of  brandy. 

14.  There  are  two  numbers  which  are  to  each  other  as 
3  to  2.     If  6  be  added  to  the  greater,  and  subtracted  from 

*  Bridge's-  Algebra. 


DIVISION.  20a 

the  less,  the  sum  and  remainder  will  be  to  each  other,  as  3 
to  1.     What  are  the  numbers?  Ans.  24  and  16. 

15.  There  are  two  numbers  whose  product  is  320;  and 
the  difference  of  their  cubes,  is  to  the  cube  of  their  differ- 
^ce,  as  61  to  1.     What  are  the  numbers  ? 

Ans.  20  and  16. 

16.  There  are  two  numbers,  which  are  to  each  other,  in 
the  duplicate  ratio  of  4  to  3  ;  and  24  is  a  mean  proportion- 
al between  them.     What  are  the  numbers  ? 

Ans.  32  and  18. 


•«^®^^- 


SECTION  xin. 


DIVISION  BY  COMPOUND  DIVISORS. 

Art.  400.  J.N  the  section  on  division,  the  case  in  which 
the  divisor  is  a  compound  quantity  was  omitted,  because  the 
operation,  in  most  instances,  requires  some  knowledge  of 
the  nature  of  powers  ^  a  subject  which  had  not  been  pre- 
viously explained. 

Division  by  a  compound  divisor  is  performed  by  the  fol- 
lowing rule,  which  is  substantially  the  same,  as  the  rule  for 
division  in  arithmetic  : 

To  obtain  the  first  term  of  the  quotient,  divide  the  first 
term  of  the  dividend,  by  the  first  term  of  the  divisor. 

Multiply  the  whole  divisor,  by  the  term  placed  in  the  quo- 
tient ;  subtract  the  product  from  a  part  of  the  dividend ;  and 
to  the  remainder  bring  down  as  many  of  the  following  terms, 
as  shall  be  necessary  to  continue  the  operation : 

Divide  again  by  the  first  term  of  the  divisor,  and  proceed 
as  before,  till  all  the  terms  of  the  dividend  are  brought 
^own. 


206  ALGEBRA. 

Ex.  1.  Divide  ac-\-bc:\-ad'^bd,  hy  a-\-h, 

a-{b)ac-^bc-]-ad+bd(c-}-d 

ac-i-bc,  the  first  subtrahend. 

*       *     ad-\-bd 

ac^-f  6c/,  the  second  subtrahend. 


Here  ac,  the  first  term  of  the  dividend,  is  divided  hy  a, 
the  first  tenn  of  the  divisor,  (Art.  116.)  which  gives  c  for  the 
firet  term  of  the  quotient.  Multiplying  the  whole  divisor  by 
this,  we  have  ac+bc  to  be  subtracted  from  the  two  first 
terms  of  the  dividend.  The  two  remaining  terms  are  then 
brought  down,  and  the  first  of  them  is  divided  by  the  first 
term  of  the  divisor,  as  before.  This  gives  d  for  tihe  second 
term  of  the  quotient.  Then  multiplying  the  divisor  by  d, 
we  have  ad-\-bd  to  be  subtracted,  which  exhausts  the  whole 
dividend,  without  leaving  any  remainder. 

The  rule  is  founded  on  this  principle,  that  the  product  of 
the  divisor  into  the  several  parts  of  the  quotient,  is  equal  to 
the  dividend.  (Art.  115.)  Now  by  the  operation,  the  pro- 
duct of  the  divisor  into  the  Jirst  term  orthe  quotient  is  sub- 
tracted from  the  dividend  ;  then  the  product  of  the  divisor 
m  the  second  term  of  the  quotient ;  and  so  on,  till  the  pro- 
duct of  the  divisor  into  each  term  of  the  quotient,  that  is, 
the  product  of  the  divisor  into  the  whole  quotient,  (Art.  100.) 
is  taken  from  the  dividend.  If  there  is  no  remainder,  it  is 
evident  that  this  product  is  equal  to  the  dividend.  If  there 
is  a  remainder,  the  product  of  the  divisor  and  quotient  is 
equal  to  the  whole  of  the  dividend  except  the  remainder. 
And  this  remainder  is  not  included  in  the  parts  subtracted 
from  the  dividend,  by  operating  according  to  the  rule. 

401.  Before  beginning  to  divide,  it  will  generally  be  ex- 
pedient to  make  some  preparation  in  the  arrangement  of  the 
ierms. 

The  letter  which  is  in  the  first  term  of  the  divisor,  should 
be  in  the  first  term  of  the  dividend  also.  And  the  powers  of 
this  letter  should  be  arranged  in  order,  both  in  the  divisor 
and  in  the  dividend;  the  highest  power  standing  first,  the 
next  highest  next,  and  so  on. 

Ex.  2.  Divide  ^an  +  b^±2ab^ +  u^,hy  a^ +b^ -^ab. 

Here  if  we  take  «*  for  the  first  term  of  the  divisor,  the 


DIVISION.  5^07 

other  terms  should  be  arranged  according  to  th«  powers  of 
05,  thus, 

In  these  operations,  particular  care  will  be  necessary  In 
the  management  of  negative  quantities.  Constant  attention 
must  be  paid  to  the  rules  for  the  signs  in  subtraction,  multi- 
plication and  division.  (Arts.  82,  105,  123.) 

Ex.  3.  Divide  ^ax-'2a^x—3a^jcy-\-Qa^x-\'axy—xy,  by 
2a -y. 

If  the  terms  be  arranged  according  to  the  powers  of  a, 
they  will  stand  thus  ; 

2a  —  y)6a^x—3a^xy—2a*X'{-axy-{-2ax-^xy{3a^X''asQ+x. 
6a^x  —  3a^xy 

*  *     —2a^x-\-axi^ 

—  2a^x-\-axy 


*  *  ■\'2ax—xy 

-\-2ax—xy 

402.  In  multiplication,  some  of  the  terms,  by  balancing 
each  other,  may  be  lost  in  the  product.  (Art.  110.)  These 
may  re-appear  in  division,  so  as  to  present  terms  in  the 
course  of  the  process,  different  from  any  which  are  in  the 
dividend. 

Ex.  4. 

a+ap)a^  ■\-x^{a^  —ax-\-x^ 
a^-\-a^x 


*   —a^x-\-x^ 


^a^x—ax^ 


ax^-^x^ 


208  ALGEBRA. 

Ex.   5. 

a*— 2a3a;+2a2a;2 


+  2a^x—Aa^x^  +  4ax^ 

*       +2a^x^-4ax^+4:r* 
+  2a2;c^— 4aa?3-f4j:* 


If  the  learner  will  take  the  trouble  to  multiply  the  quo- 
tient into  the  dirisor,  in  the  two  last  examples,  he  will  find, 
in  the  partial  products,  the  several  terms  which  appear  in  the 
process  of  dividing.  But  most  of  them,  by  balancing  each 
other,  are  lost  in  the  general  product. 

Ex.  6.  Divide  a^+a^  +  a^'b-^-ab  +  Sac+Sc,  by  a+1. 

Quotient.    a^-^-ab  +  Sc. 

Ex.7.  Divide  a-\'b-'C—ax-'bx-{-cx,hja+b^c, 

Quotient.     1-— a;. 

Ex.  8.  Divide    ^a'^  —13a^x+lla^x^  —Sax^ -\-2x^,  hj 
^a^—ax+x^.  Quotient.    a^—6ax+2x^» 

403.  When  there  is  a  remainder  after  all  the  terms  of  the 
dividend  have  been  brought  down,  this  may  be  placed  ove^ 
the  divisor  and  added  to  the  quotient,  as  in  arithmetic. 

Ex.  9. 

X 

u-{-b)ac-^bc'\-ad+bdi-x{c+d+—TT. 
ac-{-bc 


ad-\-bd 
ad+bd 


DIVISION.  209 

Ex.  10. 


ad— ah 


hd-hh 
hd-hh 


It  is  evident  that  a+^  is  the  quotient  belonging  to  the 
whole  of  the  dividend,  excepting  the  remainder  y.     (Art. 

y     . 
400.     And  7__7  is  the  quotient  belonging  to  this  remainder. 

(Art.  124.) 

Ex.  11.  Divide   Qax'{-2xy  —  Sa'b  —  hy'\-Sac-\-cy-^h,  by 

h 
3a+y.  Quotient.    2a;— 6+c+2^_^  , 

Ex.  12.  Divide  a2i-3a2+2a6-6a— 45  +  22,  by  5-3. 

10 
Quotient.  a'+2a  — 4+t3^. 

Ex.  13.  See  art.  283. 

a-\-y/h)ac^\'C^/h^\■a^/d'\' ^/hd{C'\•^/d, 
aC'\'C-\/h 


*       *       ay/d-^'.Jhd 
a^Jd'\^^Jhd 

Ex.  14.  Divide  a+^^y+avy/y+ry,  by  «+  "/y- 

Quotient.     l+r-/y, 

15.  Divide  x^ '-Zax^-\-^a^x—a^ ,hj  x—a. 

16.  Divide  2?/3-19j/2+26?/-17,  by  y-8. 

17.  Divide  a^  — l,  by  a:  — I. 

18.  Divide  4a;*-9ic2+6a?-3,  by  2a;«  +  3a-l. 

19.  Divide  0^  + 4^25 +  35*,  by  a +25. 
^0.  Divide  »*— «2;e:a^,2^»  ;p«.ft4^|jy  ajS-^^^^^a, 

28 


SECTION  XIV. 


EVOLUTION  OF  COMPOUND  QUANTITIES. 

Art.  404.  JL  HE  roots  of  compound  quantities  may  be 
extracted  by  the  following  general  rule  : 

Af.er  arranging  the  terms  according  to  the  powers  of  ond 
of  the  letters,  so  that  the  highest  power  shall  stand  first,  the 
next  liigliest  next,  &c. 

Take  the  root  of  the  first  term,  for  the  first  term  of  the 
required  root  : 

Subtract  the  power  from  the  given  quantity,  and  divide  the 
first  term  of  the  remainder,  by  the  first  term  of  the  root  invol- 
ved to  the  next  inferiour  power,  and  multiplied,  by  the  index 
of  the  given  power ;]     the  quotient  will  be  the  next  term  of 
the  root. 

Subtract  the  power  of  the  terms  already  found  from  the  giv- 
en quaniily^  and,  using  the  same  divisor,  proceed  as  before* 

This  niie  verifies  itself.  For  the  root,  whenever  a  new 
term  is  added  to  it,  is  involved,  for  the  purpose  of  subtrac- 
ting its  power  from  the  given  quantity  :  and  when  the  power 
is  equal  to  this  quantity,  it  is  evident  the  true  root  is  found. 

Ex.  1.  Extract  the  cube  root  of 
a®,  the  first  subtrahend. 


:3«*)  *      3a*,  &:c.  the  first  remainder. 


a«  +  3a*+3a4+a%  the  2d  subtrahend. 


3a!'*)*       *     — 6a4,  &;c.  the  2d  remainder. 


a6  4-3a5 -3^4 -lla^-fGa^  4-120-8. 


i  By  the  given  power  is  meant  a  power  of  the  same  name  with  the  re- 
quired root-  As  powers  and  roots  are  correlative,  any  quantity  is  the 
square  oi'  its  square  root,  the  cube  of  its  cube  root,  &c. 


EVOLUTION.  211 

.Here  a^,  the  cube  root  of  ««,  is  taken  for  the  first  term  of 
the  required  root.  The  power  a^  is  subtracted  from  the 
given  quantity.  For  a  divisor,  the  first  term  of  the  root  is 
squared,  that  is,  raised  to  the  next  inferiour  power,  and  mul- 
tiplied by  3,  the  index  of  the  given  power. 

By  this,  the  first  term  of  the  remainder  3«*.  &c.  is  divided, 
and  the  quotient  a  is  added  to  the  root.  Then  77^+ «,  the 
part  of  the  root  now  found,  is  involved  to  the  cube,  for  the 
second  subtrahend,  which  is  subtracted  from  the  whole  of 
the  given  quantity.  The  first  term  of  the  remainder  --6«4^ 
&;c.  is  divided  by  the  divisor  used  above,  and  the  quotient 
—2  is  added  to  the  root.  Lastly,  the  whole  root  is  involved 
to  the  cube,  and  the  power  is  found  to  be  exactly  equal  to 
the  given  quantity. 

It  is  not  necessary  to  write  the  remainders  at  length,  as, 
in  dividing,  the  first  term  only  is  wanted. 

2.  Extract  the  fourth  root  of 

a*-{-8a»+24a!»  +  32a-{-16(a+2 


\a^Y     8a%  &c. 

«*  +  8a^  +  24a^  4-32a-;-16. 


3.  What  is  the  5th  root  of 

Ans.   a-\-h. 

4.  What  is  the  cube  root  of 

a3-.6«2  6+12a62__863  ?  Ans.  n-2h. 

^)»  What  is  the  square  root  of 

4a*-12«6  +  9^*-f  16a/i-24M-|-16A^(2«~36-f  4/i 
4a  2 


4a)*   -12ai,  &c. 

4a2  — 12a6  +  96^ 
4a)  *         *         *     i6a/i,  &c. 

4a2--12a&-|.962  +  lGaA-246A+lGA^ 


212  ALGEBRA. 

In  finding  the  divisor  here,  the  term  2a  in  the  root  is  not 
involved,  because  the  power  next  below  the  square  is  the 
first  power. 

405.  But  the  square  root  is  more  commonly  extracted  by 
■the  following  rule^  which  is  of  the  same  nature,  as  that  which 
is  used  in  arithmetic. 

After  arranging  the  terms  according  to  tlie  powers  of  one 
of  the  letters,  take  the  root  of  the  lirst  term,  for  the  first 
term  of  the  required  root,  and  subtract  the  power  from  the 
given  quantity. 

Bring  down  two  other  terms  for  a  dividend.  Divide  by 
double  the  root  already  found,  and  add  the  quotient,  both  to 
the  root,  and  to  the  divisor.  Multiply  the  divisor  thus  in- 
creased, mto  the  term  last  placed  in  the  loot,  and  subtract 
the  product  from  the  dividend. . 

Bring  down  two  or  three  additional  terms,  and  proceed 
as  before. 

Ex.   1.  What  is  the  square  root  of 
a^,  the  first  subtrahend. 


Into  b=     2fif 6  +  6  %  the  2d  subtrahend. 


2a-\-2b+c)    *      *  2ac+2bc+c'' 

Into      c=  2«c  +  25c -j-c^,  the  3d  subtrahend. 


Here  it  will  be  seen,  that  the  several  subtrahends  are  suc- 
cessively taken  from  the  given  quantity,  till  it  is  exhausted. 
If  then,  these  subtrahends  are  together  equal  to  the  square 
of  the  terms  placed  in  the  root,  the  root  is  truly  assigned 
by  the  rule. 

The  Jirst  subtrahend  is  the  square  of  the  first  term  of  the 
root. 

The  second  subtrahend  is  the  product  of  the  second  term 
of  the  root,  into  itself,  and  into  twice  the  preceding  term. 

The  third  subtrahend  is  the  product  of  the  third  term  of 
the  root,  into  itself,  and  into  twice  the  sum  of  the  two  pre- 
ceding terms,  &c. 

That  is,  the  subtrahends  are  equal  to 

G"  +  (2«+6)  xb-{-  (2a+%  +  c)xc,  &c. 
and  this  expression  is  equal  to  the  square  of  the  root. 


EVOLUTION.  21 S 

For  {a  +  hy  =a^+2ah'\'b^  =«»  4.(2a+5)  x b,  (Art.  120.) 
And  putting  A =01  + 6,  the  square  h^  =a* -\-(9a-\-b)xh» 
And  {a-^b  +  cy={h-{-cy  =h^+(2h^ c)xc', 
-ihat  isj  restoring  the  values  of  h  and  A^, 

{a^-b-hcf  =a^+{'2a+b)xb  +  {2a  +  2b  +  c)x€. 

In  the  same  manner  it  may  be  proved,  that,  if  another 
term  he  added  to  the  root,  the  power  Avill  be  increased,  by 
the  product  of  that  term,  into  itself,  and  into  twice  the  surrl 
of  the  preceding  terms. 

The  demonstration  will  be  substantially  the  same,  if  some 
of  the  terms  be  7iegative, 

2.  What  is  the  square  root  of 

1  -46  +  462  ^^y-iby+y^  (1  -2b+y 
1 


2-26)  * -464-46' 
Into  -  26 =—46  4- 46^ 


2-46+3/)  *       *     2y-Aby+y^ 
Into  2/=  2y  —  Ahy-\-y^ 


3.  "VVliat  is  the  square  root  of 

a«— 2^5 +3a*— 2a3  +  a^  ?  Ans.     a' —  «* +«. 

4.  What  is  the  square  root  of 

«4+4a2  6+462-4a2-86+4  ?       Ans.  «'+26-2. 

406.  It  will  frequently  facilitate  the  extraction  of  roots, 
to  consider  the  index  as  composed  of  two  or  more  factors. 

Thus  a^=a*  ^  ^  (Art.  258.)  And  a*=a^~  ^  *.      That  is, 

The  fourth  root  is  equal  to  the  square  root  of  the  square 
root ; 

The  sixth  root  is  equal  to  the  square  root  of  the  cube 
root ; 

The  eighth  root  is  equal  to  the  square  root  of  the  fourth 
root,  &c. 

To  find  the  sixth  root,  therefore,  we  may  first  extract  the 
cube  root,  and  then  the  square  root  of  this. 


214  ALGEBRA. 

1.  Find  the  square  root  of  a:*— 4x^+G;c~--4a:^-l.. 
2.  Find  the  cube  root  of  ;v«-- 5a:' fl5J^4-20a; '4-1 5a;--Ga: 4-1 

3.  Find  the  square  root  of  4x^ —ix'^ -\-13^^ —Gx  +  d, 

4.  Find  the  fourth  root  of 

16a*— 96«^:c 4-216^2  x^  — 216ax^4-81x*. 

5.  Find  the  5th  root  of  a:^ -\. 5a:* +  I0x^ +  I0x^ +5x4^1. 

6.  Find  the  sixth  root  of 


^^®^>- 


SECTION  XV. 


INFINITE  SERIES. 


Art.  407.  JLT  is  frequently  the  case,  that,  in  attempting  to 
extract  the  root  of  a  quantity,  or  to  divide  one  quantity  by 
another,  we  find  it  impossible  to  assign  the  quotient  or  root 
with  exactness.  But,  by  continuing  the  operation,  one  term 
after  another  may  be  added,  so  as  to  bring  the  result  nearer 
and  nearer  to  the  value  required.  When  the  number  of 
terms  is  supposed  to  be  extended  beyond  any  determinate 
limits,  the  expression  is  called  an  infinite  series.  The  quan- 
tity^  however,  may  be  finite,  though  the  number  of  terms  be 
unlimited. 

A  fraction  may  often  be  expanded  into  an  infinite  series, 
hy  dividing  the  numerator  by  the  denominator.  For  the  -value 
of  a  fraction  is  equal  to  the  quotient  of  the  numerator  divi- 
ded by  the  denominator.  (Art.  135.)  When  this  quotient 
can  not  be  expressed,  in  a  limited  number  of  terms,  it  may 
be  represented  by  an  infinite  series. 

I 

Ex.  1 .  To  redace  the  fraction  \ to  an  infinite  series, 

1  —  a 

divide  1  by  1  —a,  according  to  th3  rule  in  art.  400. 


INFINITE  SERIES.     ,  215 

l-a)l       (l+a4-a*4-«',  &:c. 
1-a 


* 

a 

a- 

-a2 

* 

■a^ 

* 

as 

&c. 

By  continuing  the  operation,  we  obtain  the  terms 

l+Q-i-«*+a^H-a*4-a*-f  a°,  &;c.  which  are  sufficient 
to  show  that  the  series,  after  the  first  term,  consists  of  the 
powers  of  a,  rising  regularly  one  above  another, 

1 

2.  If  T-r-  be  expanded,  the  series  will  be  the  same  as  that 

from  73~5  except  that  the  terms  which  consist  of  the  odd 

powers  of  a  will  be  negative. 
1 
So  that  J— ==i_a-|-a2--a3^ct*---a5+a«  ^c. 

h  :      .         . 

3.  Reduce        ,  to  an  infinite  series. 

\  ih      bh     h^h 

a 


I  ^^^E  X^  v«  ^^  VA  V 


bh      h% 
a""  a^ 

-,-&c. 

.         h 
Here  A  divided  by  a,  gives  —   for  the  first  term  of  the 

quotient.    (Art.  124.)    This  is  multipUed  into  a— 6,  and  the 


^Ig  ALGEBRA. 

hh 

product  is /t—~  ;  (Arts.  159,  158.)  which  subtracted  froaai 

hh  hh 

h  leaves  — .     This  divided  bj  a,  gives  -r  (Art.  163.)  for  the 

second  term  of  the  quotient.      Tf  the  operation  be   con- 
tijiued  in  the  same  manner,  we  shall  obtain  the  series, 

h      hh     V^h     h^h     h'h 

in  which  the  exponents  of  h  and  of  a  increase  regularly  by  1 . 

i4.  Reduce  :;^ to  an  infinite  series. 

1— a 

Ans.    l+2a+2a^+2a3  4.2a*  (Src. 

408.   An  infinite  series  may  be  produced,  hy  extracting  the 
root  of  a  compound  surd. 


dEx.  1 .    Reduce    Va^  +  b^  to  an  infinite  series,  by  extrac- 
ting the  square  root,  according  to  the  rule  in  art.  405. 

f       h^       b*         b'' 


h^\ 


h'+-,. 


4a^ 


2a+--^r;-4^F&c. 

h^       V        l^ 


""8a^ 


2.    Va^„6^=a-^^-5^-— &c. 


3.  /SrsVl  +  Jsrsl-i-i-^+^V&C. 


XOTES. 


[Notes  A  and  B  are  omilted  in  the  abridgment.] 

Note  C.  Page  32. 

It  is  common  to  define  multiplication,  by  saying  that  '  it  is 
finding  a  product  which  has  the  same  ratio  to  the  multipli- 
cand, that  the  multiplier  has  to  a  unit.'  This  is  strictly  and 
universally  true.  But  the  objection  to  it,  as  a  definition,  is, 
that  the  idea  of  ratio,  as  the  term  is  understood  in  arithme- 
tic and  algebra,  seems  to  imply  a.  previous  knowledge  of  mul- 
tiplication, as  well  as  of  division.  In  this  work  at  least,  ge- 
ometrical ratio  is  made  to  depend  on  division,  and  division, 
on  multiplication.  Ratio,  tnerefore,  could  not  be  properly 
introduced  into  the  definition  of  multiplication. 

[Note  D  omitted  in  the  abridgment.] 

Note  E.  p.  86. 

As  the  direct  powers  of  an  integral  quantity  have  positive 
indices,  while  the  reciprocal  powers  have  negative  indices  ; 
it  is  common  to  call  the  former  positive  powers,  and  the  lat- 
ter negative  powers.  But  this  language  is  ambiguous,  and 
may  lead  to  mistake.  For  the  same  terms  are  apphed  to 
powers  with  positive  and  negative  signs  prefixed.  Thus 
-f  8a*  is  called  a  positive  power;  while  ^—a*  is  called  a 
negative  one.  It  may  occasion  perplexity,  to  speak  of  the 
latter  as  being  both  positive  and  negative  at  the  same  time  ; 
positive,  because  it  has  a  positive  index,  and  negative,  be- 
cause it  has  a  negative  co-eflicient.  This  ambiguity  may  be 
avoided,  by  using  the  terms  direct  and  reciprocal  ;  meaning, 
by  the  former,  powers  with  positive  exponents,  and,  by  the 
latter,  powers  with  negative  exponents. 

[Note  F  omitted.] 

Note  G.  p.  146. 

Every  affected  quadratic  equation  may  be  reduced  to  one 
of  the  three  following  forms. 


2 

3 
29 


1.  a;2-faa;=     h) 

2.  x^'-ax=:     h\ 

3.  ar2— «;v=— 6) 


218  ALGEBRA. 

These,  when  they  are  resolved,  become 
1.  x^-^aly/la^  +b^ 

3.  x=  ^alVia^  -b  J 
In  the  two  first  of  these  forms,  the  roots  are  never  ima- 
ginary. For  the  terms  under  the  radical  sign  are  both  pos- 
itive. But,  in  the  third  form,  whenever  h  is  greater  than 
la^ ,  the  expression  {a^  ■— 6  is  negative,  and  therefore  its 
root  is  impossible. 

[Note  H  omitted.] 

Note  I.  p.  177. 

This  definition  of  compound  ratio  is  more  comprehensive 
than  the  one  which  is  given  in  Euclid.  That  is  included  in 
this,  but  is  limited  to  a  particular  case,  Avhich  is  stated  in 
art.  353.  It  may  answer  the  purposes  of  geometry,  but 
is  not  sufficiently  general  for  algebra. 

Note  K.  p.  178. 

It  is  not  denied,  that  very  respectable  writers  use  these 
terms  indiscriminately.  But  it  appears  to  be  without  any 
necessity.  The  ratio  of  6  to  2  is  3.  There  is  certainly  a 
difference  between  twice  this  ratio,  and  the  square  of  it,  that 
is,  between  twice  three,  and  the  square  of  three.  All  are 
agreed  to  call  the  latter  a  duplicate  ratio.  What  occasion  is 
there,  then,  to  apply  to  it  the  term  *douhle  also  ?  This  is 
wanted,  to  distinguish  the  other  ratio.  And  if  it  is  confined 
to  that,  it  is  used  according  to  the  common  acceptation  of 
the  word,  in  familiar  language. 

Note  L.  p.  18o. 

The  definition  here  given'  is  meant  to  be  applicable  to 
quantities  of  every  description.  The  subject  cf  proportion, 
as  it  is  treated  of  in  Euclid,  is  embarrassed  by  the  means 
which  are  taken  to  provide  for  the  case  of  incommensurable 
quantities.  But  this  difficulty  is  avoided  by  the  algebraic  no- 
tation, which  may  represent  the  ratio  even  of  incommensu- 
rables. 

Note  M.  p.  190. 

The  inversion  of  the  means  can  be  made,  with  strict  pro- 
priety, in  those  cases  only  in  which  all  the  terms  are  quanti- 


x\OTES,  219 

lies  of  the  same  kind.  For,  if  the  two  last  be  different 
from  the  two  first,  the  antecedent  of  each  couplet,  after  the 
inversion,  wilt  be  different  from  the  consequent,  and  there- 
fore, there  can  be  no  ratio  between  them.  (Art.  355.) 

This  distinction,  however,  is  of  little  importance  in  prac- 
tice. For,  when  the  several  quantities  are  expressed  in 
numbers,  there  will  always  be  a  ratio  between  the  numbers. 
And  when  two  of  them  are  to  be  multiplied  together,  it  is 
immaterial  which  is  the  multiplier,  and  which  "the  multiph- 
cand.  Thus,  in  the  Rule  of  Three  in  arithmetic,  a  change 
in  the  order  of  the  two  middle  terms  will  make  no  difference 
in  the  result. 

Note  N.  p.  197. 

The  terms  composition  and  division  arc  derived  from  ge- 
ometry, and  are  introduced  here,  because  they  are  generally 
used  by  writers  on  proportion.  But  they  are  calculated  rath- 
er to  perplex,  than  to  assist,  the  learner.  The  objection  to 
the  word  composition  is,  that  its  meaning  is  liable  to  be  mis- 
taken for  the  composition  or  compounding  of  ratios,  (Art. 
390.)  The  two  cases  are  entirely  different,  and  ought  to  be 
carefully  distinguished.  In  one,  the  terms  are  added,  in  the 
other,  they  are  multiplied  together.  The  word  compound 
has  a  similar  ambiguity  in  other  parts  of  the  mathematics. 
The  expression  a-\-h.)  in  which  a  is  added  to  6,  is  called  a 
compound  quantity.  The  fraction  |  of  |,  or  J  x  |,  in  which 
-\  is  midtiplied  into  -f,  is  called  a  compound  fraction. 

The  term  division,  as  it  is  used  here,  is  also  exceptionable. 
The  alteration  to  which  it  is  applied,  is  effected  by  subtrac- 
tion, and  has  nothing  of  the  nature  of  what  is  called  division 
in  arithmetic  and  algebra.  But  there  is  another  case,  (Art. 
392.)  totally  distinct  from  this,  in  which  the  change  in  the 
terms  of  the  proportion  is  actually  produced  by  division. 


THE  END. 


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